MAXIMAL NILPOTENT IDEALS AND ISOMORPHISMS OF LARGE SUBALGEBRAS OF NEST ALGEBRAS

Let N be a nest of projections on a Hilbert space H and T(N) be the corresponding nest algebra. Let A be a large subalgebra of T(N). It is proved that any maximal n-nilpotent ideal of A is in the form of A∩R F,where F is a finite subnest of N and R F is the Jacobson radical of T(F).Using this result can prove that two large subalgebras are isomorphic if and only if the corresponding nests are similar.

GENERALIZED DERIVATIONS ON PARABOLIC SUBALGEBRAS OF GENERAL LINEAR LIE ALGEBRAS

Let P be a parabolic subalgebra of a general linear Lie algebra gl（n,F） over a field F, where n ≥ 3, F contains at least n different elements, and char（F） ≠ 2. In this article, we prove that generalized derivations, quasiderivations, and product zero derivations of P coincide, and any generalized derivation of P is a sum of an inner derivation, a central quasiderivation, and a scalar multiplication map of P. We also show that any commuting automorphism of P is a central automorphism, and any commuting derivation of P is a central derivation.

Frattini Subalgebras and Nonimbedding Theorem of n-Lie Algebras

In this paper, we prove the nonimbedding theorem in nilpotent n-Lie algebras which is an analogue to the nonimbedding theorem of Burnsids in groups of prime power order. We also study the properties of Frattini subalgebras of n-Lie algebras over the field with characteristic zero, and prove that the Frattini subalgebra of any k-solvable （k ≥2） n-Lie algebra is zero.

PHYSICAL SUBALGEBRA CHAINS IN A SEMISIMPLE LIE ALGEBRA

Using the defining matrices of A1 in classical algebras An, Bn, Cn and Dn, deduce the embedding indices of the physical A1 algebra in classical algebras, The Ginocchio so(8) model is as an example.

On Centralizer Subalgebras of Group Algebras

Let G be a finite group, H ≤ G and R be a commutative ring with an identity 1R. Let CRG（H）={α ∈ RG｜αh = hα for all h ∈ H）, which is called the centralizer subalgebra of H in RG. Obviously, if H=G then CRG（H） is just the central subalgebra Z（RG） of RG. In this note, we show that the set of all H- conjugacy class sums of G forms an R-basis of CRG（H）. Furthermore, let N be a normal subgroup of G and γthe natural epimorphism from G to G to G/N. Then γ induces an epimorphism from RG to RG, also denoted by % We also show that if R is a field of characteristic zero, then γ induces an epimorphism from CRG（H） to CRG（H）, that is, 7（CRG（H）） = CRG（H）.

Non-linear Invertible Maps that Preserve Staircase Subalgebras

Let g be the general linear Lie algebra consisting of all n x n matrices over a field F and with the usual bracket operation {x, y} =xy - yx. An invertible map φ ： g →g is said to preserve staircase subalgebras if it maps every staircase subalgebra to some staircase subalgebra of the same dimension. In this paper, we devote to giving an explicit description on the invertible maps on g that preserve staircase subalgebras.

Linear Commuting Maps on Parabolic Subalgebras of Finite-dimensional Simple Lie Algebras

A map φ on a Lie algebra g is called to be commuting if [φ(x), x] = 0 for all x ∈ g. Let L be a finite-dimensional simple Lie algebra over an algebraically closed field F of characteristic 0, P a parabolic subalgebra of L. In this paper, we prove that a linear mapφon P is commuting if and only if φ is a scalar multiplication map on P.

(α,β)-fuzzy Subalgebras of Q-algebras

In this note by two relations belonging to (∈) and quasi-coincidence (q) between fuzzy points and fuzzy sets, the notion of (α,β)-fuzzy Q-algebras, the level Q-subalgebra is introduced where α,β are any two of {∈,q,∈∨q,∈∧q} with α≠∈∧q. Then we state and prove some theorems which determine the relationship between these notions and Q-subalgebras. The images and inverse images of (α,β)-fuzzy Q-subalgebras are defined, and how the homomorphic images and inverse images of (α,β)-fuzzy Q-subalgebra becomes (α,β)-fuzzy Q-algebras are studied.

Subalgebra and LI-Ideal of Lattice Implication Algebras

First, we reviewed the definitions of lattice implication algebras, lattice implication subalgebras, and LI-ideals, and provided an equivalent definition of LI-ideal. Then we investigated some properties of lattice implication subalgebra and U-ideal, and found the least lattice implication subalgebra. Finally, the relation between lattice implication subalgebra and LI-ideal is presented. It is proved that no LI-ideals are non-trivial lattice implication subalgebras.

The Properties of Rational Modules and Coideal Subalgebras

In this paper, we first prove that the dual module of a rational module is still a rational module, and maximum rational modules keep their exac t properties. Then, we give the Maschke theorem of the coideal subalgebra of qua si-Frobenius algebra, and give some equivalent conditions for the ideal subcoal gebra.

ANOTHER SO(7) SUBALGEBRA IN GINOCCHIO’S MODEL

There are three subalgebra chains in Ginocchlo's so(8)model.The subalgebra so(5)+su(2)and so(7)are maximum subalgebras in so(8).The subalgebra so(6)is not this case.We can find a new subalgebra so(7)between the so(8)and so(6).It is useful in the calculation at least.

INHERITANCE OF DIVISIBILITY FORMS A LARGE SUBALGEBRA

Let A be an infinite dimensional stably finite unital simple separable C^(*)-algebra.Let B■A be a stably(centrally)large subalgebra in A such that B is m-almost divisible(m-almost divisible,weakly(m,n)-divisible).Then A is 2(m+1)-almost divisible(weakly m-almost divisible,secondly weakly(m,n)-divisible).

Dynamical Soliton Wave Structures of One-Dimensional Lie Subalgebras via Group-Invariant Solutions of a Higher-Dimensional Soliton Equation with Various Applications in Ocean Physics and Mechatronics Engineering

Having realized various significant roles that higher-dimensional nonlinear partial differ-ential equations(NLPDEs)play in engineering,we analytically investigate in this paper,a higher-dimensional soliton equation,with applications particularly in ocean physics and mechatronics(electrical electronics and mechanical)engineering.Infinitesimal generators of Lie point symmetries of the equation are computed using Lie group analysis of differen-tial equations.In addition,we construct commutation as well as Lie adjoint representation tables for the nine-dimensional Lie algebra achieved.Further,a one-dimensional optimal system of Lie subalgebras is also presented for the soliton equation.This consequently enables us to generate abundant group-invariant solutions through the reduction of the understudy equation into various ordinary differential equations(ODEs).On solving the achieved nonlinear differential equations,we secure various solitonic solutions.In conse-quence,these solutions containing diverse mathematical functions furnish copious shapes of dynamical wave structures,ranging from periodic,kink and kink-shaped nanopteron,soliton(bright and dark)to breather waves with extensive wave collisions depicted.We physically interpreted the resulting soliton solutions by imploring graphical depictions in three dimensions,two dimensions and density plots.Moreover,the gained group-invariant solutions involved several arbitrary functions,thus exhibiting rich physical structures.We also implore the power series technique to solve part of the complicated differential equa-tions and give valid comments on their results.Later,we outline some applications of our results in ocean physics and mechatronics engineering.

格蕴涵代数的区间值犹豫模糊子代数

把区间值犹豫模糊集和格蕴涵代数相结合,给出格蕴涵代数的区间值犹豫模糊子代数的概念,讨论了区间值犹豫模糊子代数与其犹豫模糊子代数的关系。证明了2个区间值犹豫模糊子代数的交是区间值犹豫模糊子代数,并结合具体例子,说明2个区间值犹豫模糊子代数并不满足类似结论。

On the Construction and Classification of the Common Invariant Solutions for Some P(1,4) -Invariant Partial Differential Equations

We consider the following (1 + 3)-dimensional P(1,4)-invariant partial differential equations (PDEs): the Eikonal equation, the Euler-Lagrange-Born-Infeld equation, the homogeneous Monge-Ampère equation, the inhomogeneous Monge-Ampère equation. The purpose of this paper is to construct and classify the common invariant solutions for those equations. For this aim, we have used the results concerning construction and classification of invariant solutions for the (1 + 3)-dimensional P(1,4)-invariant Eikonal equation, since this equation is the simplest among the equations under investigation. The direct checked allowed us to conclude that the majority of invariant solutions of the (1 + 3)-dimensional Eikonal equation, obtained on the base of low-dimensional (dimL ≤ 3) nonconjugate subalgebras of the Lie algebra of the Poincaré group P(1,4), satisfy all the equations under investigation. In this paper, we present obtained common invariant solutions of the equations under study as well as the classification of those invariant solutions.

Applications of cnoidal and snoidal wave solutions via optimal system of subalgebras for a generalized extended (2+1)-D quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering

The nonlinear evolution equations have a wide range of applications,more precisely in physics,biology,chemistry and engineering fields.This domain serves as a point of interest to a large extent in the world’s mathematical community.Thus,this paper purveys an analytical study of a generalized extended(2+1)-dimensional quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering.The Lie group theory of differential equations is utilized to compute an optimal system of one dimension for the Lie algebra of the model.We further reduce the equation using the subalgebras obtained.Besides,more general solutions of the underlying equation are secured for some special cases of n with the use of extended Jacobi function expansion technique.Consequently,we secure new bounded and unbounded solutions of interest for the equation in various solitonic structures including bright,dark,periodic(cnoidal and snoidal),compact-type as well as singular solitons.The applications of cnoidal and snoidal waves of the model in oceanography and ocean engineering for the first time,are outlined with suitable diagrams.This can be of interest to oceanographers and ocean engineers for future analysis.Furthermore,to visualize the dynamics of the results found,we present the graphic display of each of the solutions.Conclusively,we construct conservation laws of the understudy equation via the application of Noether’s theorem.

布尔代数的Fuzzy子代数的直积

讨论了布尔代数的Fuzzy子代数的直积的一些性质;给出了直积布尔代数的Fuzzy子代数可分解为两个Fuzzy子代数的直积的充要条件;并且讨论了Fuzzy商布尔代数的直积特征。

一些特殊项链李代数的同态

项链李代数是新的一类无限维李代数。定义了项链字的左右指标数组,并利用左右指标数组,把N_Q的基分成5类,并重点讨论了项链李代数的同态的性质。

系统H_α中的广义重言式理论

给出了一类带参数的[0,1]上的t 模 α及与之伴随的新的蕴涵算子Hα(0≤α≤1),进而建立了多值系统Hα.当α=1时,Hα就是R0型蕴涵算子;当α=0时,Hα是G del蕴涵算子,所以R0算子与G del算子在Hα系统中统一起来.在,∨,→1Hα系统中引入了带参数的非运算 α,研究了多值系统H1),以H1=( 12222为赋值域建立了F(S)中重言式的分类定理,并将广义重言式分类定理推广到系统Hα(0<α<1)中.

区间值模糊命题逻辑的最大子代数及其广义重言式

将 S-型蕴涵算子改为 R0 -蕴涵算子 ,从而找到区间值模糊逻辑 I[0 ,1]的一个最大子代数 IQ,进而将王国俊教授在逻辑系统 W中的广义重言式理论推广应用到 IQ 中。