关于半局部群环

Let R be an associative ring with identity.R is said to be semilocal if R/J(R)is(semisimple)Artinian,where J(R)denotes the Jacobson radical of R.In this paper,we give necessary and sufficient conditions for the group ring RG to be semilocal,where G is a locally finite nilpotent group.     

A counter example for nilpotent groups

Let R be a ring with 1, I be a nilpotent subring of R (there exists a natural number n, such that In = 0), and I be generated by {xj |j ∈ J} as ring. Write U = 1 + I, and it is a nilpotent group with class ≤ n - 1. Let G be the subgroup of U which is generated by {1 + xj|j ∈ J}. The group constructed in this paper indicates that the nilpotency class of G can be less than that of U.     

A Ring with Left Identities anb Right Inverses Is a Skew Field

It is well known that a semigroup in which every element has a left identity and right inverses need not be a group. In this note we prove that if the semigroup is the multiplicative semigroup of a ring, then it is a group. In fact, we have the following theorem.     

ON INNERπ'-CLOSED GROUPS AND NORMAL π-COMPLEMENTS

In this paper, the author classifies the finite inner π′-closed groups, and proves the following results1. If each proper subgroup K of a group G is weak π-homogeneous and weak π′-homogeneous, then G is a Schmidt group, or a direct product of two Hall subgroups.2. If G is a weak π-homogeneous group, then G is π′-closed if one of the following statements is true: (1)Each π-subgroup of G is 2-closed. (2) Each π-subgroup of G is 2′-closed.3. Let G be a group and π be a set of odd primes. If N_G(Z(J(P))) has a normal π-completement for a Sytow p-subgroup of G with prime ρ in π then so does G.     

A Comment on a Necessary and Sufficient Condition for Transitive Permutation Group by P.J.Cameron

P.J. Cameron had mentioned that "It can be shown that a permutation group is transitive if and only if its centralizer in the symmetric group is semiregular, and vice versa (Wielandt, page 9)."[1] The latter is true, i.e. G≤S_Ω is semiregular(?)Cs_Ω(G) is transitive. [cf. 2] But in the former statement fails, i.e. Cs_Ω(G) is semiregular(?)G≤S_Ω is transitive. In this note we give a series of the     

COMPENSATED COMPACTNESS AND THE STRATIFIED LIE GROUP

In this paper we prove that the Jacobian J(F) of a map F( f1, , fl) from Ginto Rl maps the product of Lebesgue space Lp1 × × Lpl into local Hardy space hγ(G),where Q/(Q+1)<γ≤ 1, and Q is the homogeneous dimension of the stratified Lie group G .     

TOEPLITZ OPERATORS AND ALGEBRAS ON DIRICHLET SPACES

The automorphism group of the Toeplitz C-algebra,J(C~1),generated by Toeplitz op-erators with C~1-symbols on Dirichlet space D is discussed;the K_0,X_1-groups and the firstcohomology group of J(C~1)are computed.In addition,the author provs that the spectraof Toeplitz operators with C~1-symbols are always connected,and discusses the algebraic prop-erties of Toeplitz operators.In particular,it is proved that there is no nontrivial selfadjointToeplitz operator on D and T_φ~*=T_φ if and only if T_φ is a scalar operator.     

群逆的仿射组合表示

In this paper, we first give two equalities in the operation of determinant. Using the expression of group inverse with full-rank factorization Ag = F（GF）^-2G and the Cramer rule of the nonsingular linear system Ax = b, we present a new method to prove the representation of group inverse with arlene combination
Ag=∑（I,J）∈N（A） 1/υ^2det（A）IJ ajd AJI.
A numerical example is given to demonstrate that the formula is efficient.     

L—Fuzzy群（I）

In this paper,the L-fuzzy group theory in a general frame is established,which is a generalization of [1-4].     

L—fuzzy群（Ⅱ）

This paper is a continuation of [1].In paper[1],the concept of LF group was introduced.In this paper,the concept of OP homomorphism will be introduced and its properties will be discussed.     

On locally spherical polytopes of type {5, 3, 5}

There are only finitely many locally projective regular polytopes of type {5, 3, 5}. They are covered by a locally spherical polytope whose automorphism group is J₁×J₁×L₂(19), where J₁ is the first Janko group, of order 175560, and L₂(19) is the projective special linear group of order 3420. This polytope is minimal, in the sense that any other polytope that covers all locally projective polytopes of type {5, 3, 5} must in turn cover this one.     

Cohomology for infinitesimal unipotent algebraic and quantum groups

In this paper we study the structure of cohomology spaces for the Frobenius kernels of unipotent and parabolic algebraic group schemes and of their quantum analogs. Given a simple algebraic group G, a parabolic subgroup P _J , and its unipotent radical U _J , we determine the ring structure of the cohomology ring H^?((U _J )₁, k). We also obtain new results on computing H^?((P _J )₁, L(??)) as an L _J -module where L(??) is a simple G-module with highest weight ?? in the closure of the bottom p-alcove. Finally, we provide generalizations of all our results to small quantum groups at a root of unity.     

The Kernel of the Eisenstein Ideal

Let N be a prime number, and let J₀(N) be the Jacobian of the modular curve X₀(N). Let T denote the endomorphism ring of J₀(N). In a seminal 1977 article, B. Mazur introduced and studied an important ideal I⊆T, the Eisenstein ideal. In this paper we give an explicit construction of the kernel J₀(N)[I] of this ideal (the set of points in J₀(N) that are annihilated by all elements of I). We use this construction to determine the action of the group Gal(Q/Q) on J₀(N)[I]. Our results were previously known in the special case where N−1 is not divisible by 16.     

Polynomial identities for the Jordan algebra of a degenerate symmetric bilinear form

Let _Jn$J_n$ be the Jordan algebra of a degenerate symmetric bilinear form. In the first section we classify all possible G  -gradings on _Jn$J_n$ where G   is any group, while in the second part we restrict our attention to a degenerate symmetric bilinear form of rank n−1$n-1$, where n is the dimension of the vector space V   defining _Jn$J_n$. We prove that in this case the algebra _Jn$J_n$ is PI-equivalent to the Jordan algebra of a nondegenerate bilinear form.     

Some results on the capitulation problem

Let K be an unramified abelian extension of a number field F with Galois group G. K corresponds to a subgroup H of the ideal class group of F. We study the subgroup J of ideal classes in H which become trivial in K. There is an epimorphism from the cohomology group H^?1(G, Cl_K) to J which is an isomorphism if G is cyclic; Cl_K is the ideal class group of K. Some results on the structure of J and Cl_K are obtained.     

On Jacobi functions and multiplication theorems for integrals of Bessel functions

In this paper, the product of a Jacobi polynomial and function is shown to generate the Jacobi polynomial. This type of expansion was previously known for all the classical orthogonal polynomials except the Jacobi. The result is then used to obtain generalizations to Watson's [10] multiplication theorem involving integrals of Bessel functions. The integrals considered are of the form ∝^∞₀t^1?λJ_v(tx₁) J_μ(tx₂) J_σ(tx₃) J_τ(tx₄) dt.     

Stopping times and an extension of stochastic integrals in the plane

Let (Ω,$\text{J}$,P;$\text{J}$_z) be a probability space with an increasing family of sub-σ-fields {$\text{J}$_z, z ∈ D}, where D = [0, ∞) × [0, ∞), satisfying the usual conditions. In this paper, the stochastic integral with respect to an $\text{J}$_z-adapted 2-parameter Brownian motion for integrand processes in the class $\text{C}$₂($\text{J}$_z) is extended, by means of truncations cations by {0, 1}-valued 2-parameter stopping times, to integrand processes that are $\text{J}$_z-adapted and continuous. The stochastic integral in the plane thus extended resembles a locally square integrable martingale in the 1-parameter setting. A definition of a parameter-space valued, i.e., D-valued, stopping time is also given and its characteristic process is related to a {0, 1}-valued 2-parameter stopping time.     

Bounds for the first positive zero of a mixed Bessel function

The present article is concerned with lower and upper bounds of the first positive zero of the function H_ν(z, α) = αJ_ν(z) + zJ′_ν(z), Where J_gn(z) is the ordinary Bessel function of order ν > −1 and J′_ν(z) is the derivative of J_ν(z). A lower bound found here improves and extends the range of validity of the order ν, of a lower bound found in a previous work [8]. Also, two upper bounds given here improve a previously known upper bound [8]. In the particular case α = 0, these bounds lead to lower and upper bounds for the first positive zero j′_ν,1 of J′_ν(z) which improve well-known bounds in the literature.     

On invisible elements of the Tate-Shafarevich group

Mazur [7] has introduced the concept of visible elements in the Tate-Shafarevich group of optimal modular elliptic curves. We generalized the notion to arbitrary abelian subvarieties of abelian varieties and found, based on calculations that assume the Birch-Swinnerton-Dyer conjecture, that there are elements of the Tate-Shafarevich group of certain sub-abelian varieties of J₀ (p) and J₁ (p) that are not visible.     

Extremes of conditioned elliptical random vectors

Let {X_n,n?1} be iid elliptical random vectors in R^d,d≥2 and let I,J be two non-empty disjoint index sets. Denote by X_n,I,X_n,J the subvectors of X_n with indices in I,J, respectively. For any a∈R^d such that a_J is in the support of X_1,J the conditional random sample X_n,I|X_n,J=a_J,n≥1 consists of elliptically distributed random vectors. In this paper we investigate the relation between the asymptotic behaviour of the multivariate extremes of the conditional sample and the unconditional one. We show that the asymptotic behaviour of the multivariate extremes of both samples is the same, provided that the associated random radius of X₁ has distribution function in the max-domain of attraction of a univariate extreme value distribution.