Metric n-Lie Algebras

An n-Lie algebra is said to be metric if it is endowed with an invariant, non-degenerate, symmetric bilinear form. We prove that any simple n-Lie algebra over an algebraically closed field of characteristic zero admits a unique metric structure and vice versa. Further, we present two metric n-Lie algebras, which are indecomposable but admit many more metric structures.     

Isoclinism between pairs of n-Lie algebras and their properties

In this paper, we investigate the notion of isoclinism on a pair of n-Lie algebras, which forms an equivalence relation. In addition, we prove that each equivalence class contains a stem pair of n-Lie algebras, which has minimal dimension amongst the finite dimensional pairs of n-Lie algebras. Finally, some more results are obtained when two isoclinic pairs of n-Lie algebras are given.     

Some Properties On Isoclinism in n-Lie Algebras

The concept of n-Lie algebra were introduced by Filippov in 1987. One notes that not all properties of Lie algebras can be carried over to n-Lie algebras, as Williams showed in 2009. In the present article, among other results it is shown that the notions of isoclinism and isomorphism for two finite dimensional n-Lie algebras of the same dimension are equivalent. This was already done for ordinary Lie algebras by Moneyhun in 1994.     

基本n-Lie代数

In this paper, we mainly study some properties of elementary n-Lie algebras, and prove some necessary and sufficient conditions for elementary n-Lie algebras. We also give the relations between elementary n-algebras and E-algebras.     

特征幂零n-Lie代数

研究了一类特殊的幂零n-Lie代数,其所有导子都是幂零的,称其为特征幂零n-Lie代数.本文主要是得到了一般的n-Lie代数是特征幂零n-Lie代数的一些充要条件.     

n+1维n-Lei代数的导子代数

本文给出了n 1维n-Lie代数的导子代数的结构，以及导子的具体表示形式．     

On the Constant Metric Dimension of Generalized Petersen Graphs P（n, 4）

In this paper,we consider the family of generalized Petersen graphs P(n,4).We prove that the metric dimension of P(n,4) is 3 when n ≡ 0(mod 4),and is 4 when n = 4k + 3(k is even).For n ≡ 1,2(mod 4) and n = 4k + 3(k is odd),we prove that the metric dimension of P(n,4) is bounded above by 4.This shows that each graph of the family of generalized Petersen graphs P(n,4)has constant metric dimension.     

n-Lie代数的Frattini子代数及非嵌入定理

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

On the metric dimension of HDN

The concept of metric basis is useful for robot navigation. In graph G, a robot is aware of its current location by sending signals to obtain the distances between itself and the landmarks in G. Its position is determined uniquely in G if it knows its distances to sufficiently many landmarks. The metric basis of G is defined as the minimum set of landmarks such that all other vertices in G can be uniquely determined and the metric dimension of G is defined as the cardinality of the minimum set of landmarks. The major contribution of this paper is that we have partly solved the open problem proposed by Manuel et al. [9], by proving that the metric dimension of $\text{HDN1}(n)$ and $\text{HDN2}(n)$ are either 3 or 4. However, the problem of finding the exact metric dimension of HDN networks is still open.     

单的n+1维n-李代数的表示

本文证明了不可约的L(A)-模是A-模的充要条件,给出了单的n 1-维n-李代数的有限维不可约表示的分类．     

Some results on monotone metric spaces

We present some new results on monotone metric spaces. We prove that every bounded 1-monotone metric space in R^d${\mathbb{R}}^d$ has a finite 1-dimensional Hausdorff measure. As a consequence we obtain that each continuous bounded curve in R^d${\mathbb{R}}^d$ has a finite length if and only if it can be written as a finite sum of 1-monotone continuous bounded curves. Next we construct a continuous function f such that M   has a zero Lebesgue measure provided the graph(f|M)$\mathrm{graph}(f|M)$ is a monotone set in the plane. We finally construct a differentiable function with a monotone graph and unbounded variation.     

The Decomposition and uniqueness of n-Lie Superalgebras

In this paper, the definitions and some properties of n-Lie superalgebras are presented. Our main aim is to study the decomposition and uniqueness of finite dimensional n-Lie superalgebras with trivial center. Aecoding to the decomposition of n-Lie superalgebras, we obtain the decomposition of inner derivation superalgebras and derivation superalgebras respectively. Furthermore, we discuss some properties about the centroid of n-Lie superalgebras, so we can see its application in the decomposition of n-Lie superalgebras.     

Some periodic point results in generalized metric spaces

We prove some fixed point and periodic point theorems for a map in generalized metric spaces. An example is provided to support our result. The results presented in this paper generalize several well known comparable results in the literature.     

Some G-M-type Banach spaces and K-groups of operator algebras on them

By providing several new varieties of G-M-type Banach spaces according todecomposable and compoundable properties, this paper discusses the operator structuresof these spaces and the K-theory of the algebra of the operators on these G-M-type Ba-nach spaces through calculation of the K-groups of the operator ideals contained in theclass of Riesz operators.     

Some Properties of a Class Self—homomorphism of BCI—algebras

ＢｙａＢＣＩ－ａｌｇｅｂｒａｗｅｍｅａｎａｎａｌｇｅｂｒａ（Ｘ；，０）ｏｆｔｙｐｅ（２，０）ｓａｔｉｓｆｙｉｎｇｔｈｅａｘｉｏｍｓ：（１）（（ｘｙ）（ｘｚ））（ｚｙ）＝０；（２）（ｘ（ｘｙ））ｙ＝０；（３）ｘｘ＝０；（４）ｘｙ＝ｙｘ＝０ｘ＝ｙｆｏｒａｎｙｘ，ｙａｎｄｚｉｎＸ．ＦｏｒａｎｙＢＣＩ－ａｌｇｅｂｒａＸ，ｔｈｅｒｅｌａｔｉｏｎ≤ｄｅｆｉｎｅｄｂｙｘ≤ｙｉｆａｎｄｏｎｌｙｉｆｘｙ＝０ｉｓａｐａｒｔｉａｌｏｒｄｅｒｏｎＸ［１］．ＩｎａｎｙＢＣＩ－ａｌｇｅｂｒａＸ，…     

The metric dimension of Cayley digraphs

A vertex x in a digraph D is said to resolve a pair u, v of vertices of D if the distance from u to x does not equal the distance from v to x. A set S of vertices of D is a resolving set for D if every pair of vertices of D is resolved by some vertex of S. The smallest cardinality of a resolving set for D, denoted by dim(D), is called the metric dimension for D. Sharp upper and lower bounds for the metric dimension of the Cayley digraphs Cay(Δ:Γ), where Γ is the group Z_n1⊕Z_n2⊕?⊕Z_nm and Δ is the canonical set of generators, are established. The exact value for the metric dimension of Cay({(0,1),(1,0)}:Z_n⊕Z_m) is found. Moreover, the metric dimension of the Cayley digraph of the dihedral group D_n of order 2n with a minimum set of generators is established. The metric dimension of a (di)graph is formulated as an integer programme. The corresponding linear programming formulation naturally gives rise to a fractional version of the metric dimension of a (di)graph. The fractional dual implies an integer dual for the metric dimension of a (di)graph which is referred to as the metric independence of the (di)graph. The metric independence of a (di)graph is the maximum number of pairs of vertices such that no two pairs are resolved by the same vertex. The metric independence of the n-cube and the Cayley digraph Cay(Δ:D_n), where Δ is a minimum set of generators for D_n, are established.     

There do not exist minimal algebras between C*(X) and C(X) with prescribed real maximal ideal spaces

Let C(X) be the algebra of all real-valued continuous functions on a completely regular Hausdorff space X, and C*(X) the subalgebra of bounded functions. We prove that for any intermediate algebra A between C*(X) and C(X), other than C*(X), there exists a smaller intermediate algebra with the same real maximal ideals as in A. The space X is called A-compact if any real maximal ideal in A corresponds to a point in X. It follows that, for a noncompact space X, there does not exist any minimal intermediate algebra A for which A is A-compact. This completes the answer to a question raised by Redlin and Watson in 1987.     

On ideal extensions of partial monounary algebras

In the present paper we introduce the notion of an ideal of a partial monounary algebra. Further, for an ideal (I, f _I ) of a partial monounary algebra (A, f _A ) we define the quotient partial monounary algebra (A, f _A )/(I, f _I ). Let (X, f _X ), (Y, f _Y ) be partial monounary algebras. We describe all partial monounary algebras (P, f _P ) such that (X, f _X ) is an ideal of (P, f _P ) and (P, f _P )/(X, f _X ) is isomorphic to (Y, f _Y ). This work was supported by the Slovak VEGA Grant No. 1/3003/06 and by the Science and Technology Assistance Agency under the contract No. APVT-20-004104.