A NEW TYPE OF PRONORMAT SUBGROUPS IN CHEVALLEY GROUPS

51. IntroductionNotation in this paper follows [2,3]. Let L be a simple Lie algebra over an algebraicclosed field of characteristic 0,. its root system with a ford base A = {afl, or2,', al},and W the Weyl group of L. . (.--) denotes the set of all the positive (negative) roots.Each s E ac can be written piquely in the form 8 = Z aam with all the coefficients beingaamintegers. Moreover, either all ac 2 0 or all ac S 0. The sum Z ac of all these coefficielltsaamis called the height of 8, …     

THE AUTOMORPHISMS OF TWO-DIMENSIONAL LINEAR GRCUPS OVER COMMUTATIVE RINGS~*

The present paper determines the form of automorphisms of E_2(R) and GE_2(R) over commutative rings provided 2, 3 and 5 are units.     

A Note on Restricted Representations of the Witt Superalgebras

Let F be an algebraically closed field of prime characteristic p 〉 3, and W（n） the Witt superalgebra over F, which is the Lie superalgebra of superderivations of the Grassmann algebra in n indeterminates. The dimensions of simple atypical modules in the restricted supermodule category for W（n） are precisely calculated in this paper, and thereby the dimensions of all simple modules can be precisely given. Moreover, the restricted supermodule category for W（n） is proved to have one block.     

Character Formulas for a Class of Simple Restricted Modules over the Simple Lie Superalgebras of Witt Type

Let F be an algebraically closed field of prime characteristic, and W(m, n, 1) be the simple restricted Lie superalgebra of Witt type over F, which is the Lie superalgebra of superderivations of the superalgebra ■(m; 1) ■∧(n), where ■(m; 1) is the truncated polynomial algebra with m indeterminants and ∧(n) is the Grassmann algebra with n indeterminants. In this paper, the author determines the character formulas for a class of simple restricted modules of W(m, n, 1) with atypical weights of type...     

A New Type of Pronormal Subgroups in Chevalley Groups

Let L be a simple Lie algebra with irreducible root system φ having roots of different length,F be a field of characteristic different from 2,G=L(F) be a Chevalley group of type L over F.Denote by φ^1 the set of all long roots in φ.Set G^1=(zr(t);r∈φ^t,t∈F).It is a subgroup of G generated by all the long root subgroups.This paper determines the pronormality of G^1 in G when L is not of type G2.     

Upper bound of the third edge-connectivity of graphs

Let G be a simple connected graph of order n≥6. The third edge-connectivity of G is defined as the minimum cardinality over all the sets of edges, if any, whose deletion disconnects G and every component of the resulting graph has at least 3 vertices. In this paper, we first characterize those graphs whose third-edge connectivity is well defined, then establish the tight upper bound for the third edge-connectivity.     

二面体群上的Hopf Ore扩张

In this paper, the Hopf Ore extension and corresponding module extension of the group algebra over dihedral group are studied. It turns out that the 1-dimensional and 2- dimensional simple representations can both be extended to the simple representations over a class of Hopf Ore extension.     

利用有限几何构作3-PBIB（2）设计和3-设计

The concept of t-PBIB design was introduced by Wei Wandi and Yang Benfu as a generalization of G-design and PBIB design. In this paper, a number of 3-associntion schemes and 3-PBIB(2) designs are constructed on the bases of the finite vector space over Fq and the finite unitary geometry over Fq2. Then a number of 3-designs are constructed on the bases of the finite orthogonal geometries over Fq. The parameters of all these designs are computed.     

COHOMOLOGY OF GRADED LIE ALGEBRAS OF CARTAN TYPE S (n, m)

Let F be an algebraically closed field of characteristic p>3. This paper studies the cohomology of graded Lie algebras S(n,m) of Cartan type over F. The author determines the structures of H~1(S(3,m), _0), where _0 is the graded S(3,m)-module which is induced by an irreducible sl (3)-module V_0, the structures of H~1(S(3,m), V), where V is an irreducible S(3,m)-module, and the structures of the restricted cohomology H_*~1(S(3,(1, 1, 1)), V), where V is an irreducible restricted S(3, (1, 1, 1))-module.     

SENSITIVITY OF THE EIGENVALUES OF A DEFECTIVE MATRIX

This paper is concerned with the sensitivity of the eigenvalues of a defective matrixunder small perturbations.The given estimate generalizes all special results of Wilkinson,Stewart,Bauer and Fike,when the eigenvalue is simple and when the matrix is nondefective,and interpretes the phenomenon indicated by Golub and Wilkinson for Multiple eigenvalues     

BASES OF BIVARIATE SPLINE SPACES WITH CROSS-CUT GRID PARTITIONS

In this paper we determine the dimensions of certain bivariate spline spaces with cross-cut grid partitions and give explicit expressions of their bases functions. As an immediate consequence, the closures of these spaces over all grid partitions of the same type are known. The results in this paper can be applied to interpolation and approximation by bivariate spline functions.     

TRACING A PLANAR ALGEBRAIC CURVE

In this paper, an algorithm that determines a real algebraic curve is outlined. Its basicstep is to divide the plane into subdomain1s that include only simple branches of the algebraic curvewithout singular points. Each of the branches is then stably and efficiently traced in the particularsubdomain. Except for tracing, the algorithm requires only a couple of simple operations on poly-nomials that ran be carried out exacrly if the coefficients are rational, and the determination of the real roots of several univariate polynomials.     

序半环的序理想

An ordered semiring is a semiring S equipped with a partial order ≤ such that the operations are monotonic and constant 0 is the least element of S.In this paper,several notions,for example,ordered ideal,minimal ideal,and maximal ideal of an ordered semiring,simple ordered semirings,etc.,are introduced.Some properties of them are given and characterizations for minimal ideals are established.Also,the matrix semiring over an ordered semiring is consid-ered.Partial results obtained in this paper are analogous to the corresponding ones on ordered semigroups,and on the matrix semiring over a semiring.     

有限域上的重根自对偶负循环码

Let F_q be a finite field with q = p~m, where p is an odd prime. In this paper, we study the repeated-root self-dual negacyclic codes over Fq. The enumeration of such codes is investigated. We obtain all the self-dual negacyclic codes of length 2~ap~r over F_q, a ≥ 1.The construction of self-dual negacyclic codes of length 2~abp~r over F_q is also provided, where gcd(2, b) = gcd(b, p) = 1 and a ≥ 1.     

On Holomorphic Automorphisms of a Class of Non-homogeneous Rigid Hypersurfaces in C^N＋1

The author determines the real-analytic infinitesimal CR automorphisms of a class of non-homogeneous rigid hypersurfaces in C^N＋1 near the origin, and the connected component containing the identity transformation of all locally holomorphic automorphisms of these hypersurfaces near the origin.     

Characteristic Semi-simple Lie Algebras and Completely Semi-simple Lie Algebras over any Field

In this paper, we study some properties of the finite dimensional characteristic semi-simple (C.S.S.) Lie algebras and completely semi-simple Lie algebras over any field. The definitions and some results of these algebras have been given by G.B.Seligman in [1]. We can easily prove the following lemmas: Lemma 1 The centre of any finite dimensional Lie algebra L is a characteristic ideal of L.     

On reducible Heegaard splittings

Let V∪S W be a reducible Heegaard splitting of genus g = g(S)≥2.For a maximal prime connected sum decomposition of V∪S W,let q denote the number of the genus 1 Heegaard splittings of S2×S1 in the decomposition,and p the number of all other prime factors in the decomposition.The main result of the present paper is to describe the relation of p,q and dim(C V∩CW).     

PERIODIC ORBITS AND P STABLE ORBIT CLOSURES OF CONTINUOUS FLOWS ON SURFACES

In this paper the author determines the maximum number of simple closed curves with certain propertiees on any manifold M of dimension 2, and obtains some general properties of M. Then applying the properties to continuous flows defined on M, the author obtains the maximum number of distinct P stable orbit closures and non-zero-homotopic periodic orbits which are not homotopic to each other.     

A characterization of finite simple groups by the orders of solvable subgroups

Let G be a finite group and S be a finite simple group. In this paper, we prove that if G and S have the same sets of all orders of solvable subgroups, then G is isomorphic to S, or G and S are isomorphic to Bn(q), Cn(q), where n≥3 and q is odd. This gives a positive answer to the problem put forward by Abe and Iiyori.     

斜场上的某些可中心化的阵

On the basis of some results of the Xie Bangjie professor, this paper proved: Theorem I. The quadratic zero matrices, the idempotent matrices and the involutory matrices over any skew-field are all centralizable matrices, and (i) If A is a quadratic zero matrix, then ‖A‖=0;