What we want to give in this paper is when is an ideal in function algebras finitely generated in view of subelliptic estimates of ∂-Neumann problem.

Let G be a finite nonsolvable group and H a proper subgroup of G. In this paper we determine the structure of G if G satisfies one of the following conditions:
(1)Every solvable subgroup K(K H) is either p-decomposable or a Schmidt group, p being the smallest odd prime factor of |G|.
(2)|G:H|is divisible by an odd prime and every solvable subgroup K(K H) is either 2'-closed or a Schmidt group.
(3)|G:H|is even and every solvable subgroup K(K H) is either 2-closed or a Schmidt group.

A Boolean algebra B= is recursive if B is a recursive subset of ω and the operations Λ, v and ┌ are partial recursive. A subalgebra C of B is recursive an (r.e.) if C is a recursive (r.e.) subset of B. Given an r.e. subalgebra A, we say A can be split into two r.e. subalgebras A₁ and A₂ if (A₁ ∪ A₂)^*= A and A₁ ∩ A₂={0, 1}. In this paper we show that any nonrecursive r.e. subalgebra of B can be split into two nonrecursive r.e. subalgebras of B. This is a natural analogue of the Friedberg's splitting theorem in ω recursion theory.

The purpose of this paper is to obtain a formula for computing the first Cartan invariant C₁₁⁽ⁿ⁾ of the group Sp(4, pⁿ). The main results are as follows:
Theorem A.If p≥7, then where r, s are the roots of x²-24x+64.
Theorem B.If p>7, then

where a, b, c, d are the roots of x⁴ -64x³+ 804x² -2672x+2048; e, f, g. are the roots of x³-36x²+256x-512; α, β, γ are the roots of x³-43x²+312x-512.

In this paper we deal with the estimates of the order of determinacy of finitely determined map-germs. This problem has been noted by mathematicians for more than ten years. But the results given by some authors for the estimates of the order of determinacy are stated with respect to a, while those given in this paper are stated with respect to a_k, where k may be any non-negative integer, and we have a₀= a in particular.

Let X and Y denote two complex Banach spaces and let B(Y, X) denote the algebra of all bounded linear operators from Y to X. For A∈B(X)ⁿ, B∈B(Y)ⁿ, the elementary operator acting on B(Y, X) is defined by . In this paper we obtain the formulae of the spectrum and the essential spectrum of Δ(A, B) by using spectral mapping theorems. For n=1, we prove that S_p(L_A, R_B)=σ(A)×σ(B) and .

Let G be a linear algebraic group over an algebraically closed field K. We call a (rational) G-module cyclic if it is generated by one element, and call it cocyclic if its dual is cyclic. We call G a c.c. group if the cyclicity is equivalent to the cocyclicity for any G-module. If G is not a c.c. group, the critical number of G is the greatest integer c(G) such that the cyclicity is equivalent to the cocyclicity for any G-module of dimension ≤ c(G).

A relationship between the Berlekamp-Massey and the Euclidean algorithms for linear feedback shift register synthesis is established. In fact, by refining a sequence of polynomials appearing in the Euclidean algorithm, the sequence of characteristic polynomials computed by the Berlekamp-Massey algorithm will be achieved.

In the present paper a few fixed point theorems are given for upper hemi-continuous mappings from a paracompact convex set to its embracing space, a real, locally convex, Hausdorff topological vector space.

In this paper we develop a sequence Z₀, …, Z_α,… of axiom systems for set theory, such that (1) the consistency of any system within the sequence is provable in its succeeding systems, (2) the first system in the sequence is Zermelo's system Z and the union of all systems in the sequence is just ZF. And we prove that for ordinal number α>1, there exists a sequence of N_a+1 axiom systems between systems Z_α and Z_α+1 such that these systems satisfy the above condition (1).

This paper deals with the following problem: whether the quasisimilarity of subnormal operators would imply the equality of their essential spectrum. It is shown that if a subnormal operator is quasisimilar to a quasinormal operator, then they have the same essential spectrum Furthermore, if the quasinormal operator in this case is almost normal, then they are unitarily equivalent up to a compact perturbation.

Let Ω⊂Rⁿ(n≥2) be a bounded open set; Q_T=Ω×[0, T], S_T=∂Ω×[0, T], S₁, S₂ be the partial boundaries of Ω and S₁∪S₂=∂Ω, S₁∩S₂=Φ. We denote Γ_{1. T}=S₁×[0, T], Γ_{2. T}= S₂×[0, T], and consider the problem

where a_i(a, x, t, u, u_ax) is not necessarily monotonic as usually is. Some other conditions are also weakened. Under such mild conditions we obtain the existence of a generalized solution of the mixed boundary value problem for the above degenerate parabolic equation.We still employ the Galerkin method and some techniques to prove the main lemma. The method used to prove our theorem is still the one initiated by us before.Finally, an example is given to illustrate the application of our theorem in the non-steady permeating problem originating in hydrokinetics.

In this paper, we shall prove the existence of the singular directions related to Hayman's problems ^[1]. The results are as follows.Suppose that f(z) is a transcendental integral function in the finite plane, then there exists a direction H: arg z= θ₀ (0≤θ₀>2π) ε, every integer p(≠0, -1) and every finite complex number b(≠0), we have

Suppose that f(z) is a transcendental integral function in the finite plane, then there exists a direction H:z= θ₀ (0≤θ₀>2π) ε, every integrer p(≥3) and any finite complex numbers a(≠0) and b, we have

Suppose that f(z) is a meromorphic function in the finite plane and satisfies the following condition

then there exists a direction H:z= θ₀ (0≤θ₀>2π) ε, every integer p(≥5) and every two finite complex numbers a(≠0) and b, we have

The singular directions in Theorems Ⅰ-Ⅲ are called Hayman directions.

We will consider a C^r diffeomorphism of the real line R, and give a necessary and sufficient condition for a C^r diffeomorphism of R to be embedded (uniquely) in a C^r flow. As an application, we do the same for diffeomorphisms of the circle S¹ and a class of analytic diffeomorphisms of the plane R².

Hirsch [1,2] studied the limiting behavior of solutions of competitive or cooperative systems, and showed that if L is an ω-limit set of a three-dimensional cooperative system, which contains no equilibrium, then L is a nonattracting closed orbit. Smith ^[3] considered a three-dimensional irreducible competitive system and showed that an ω-limit set containing no equilibrium must be a closed orbit which has a simple Floquet multiplier λ<1, and may be attracting. In this paper we carry out the qualitative analysis of a class of competitive and cooperative systems, and a generalization of the result of Levine ^[4] is given. The stability problem of closed orbits raised in [5] and [6] is resolved.

In this paper we analyze globally the behavior of the solutions of a class of cooperative systems. Our main results is that every orbit of the cooperative system (3.1) either approaches the equilibrium (0, 0, 0), or is unbounded, as t→+∞.

To study the uniform behavior of solutions to functional differential equations with infinite delay, this paper introduces a new space C_g of the initial functions with a new norm |·|_g. Corresponding definitions of g-uniform boundedness (g-UB) and g-uniform ultimate boundedness (g-UUB), and theorems ensuring g-UB org-UUB are given.

This paper is devoted to the ergodicity of generalized long-range exclusion processes with positive recurrent transition probabilities. The set of invariant probability measures and the corresponding domain of attraction for each invariant probability measure are described.

Three dimensional initial boundary value problem of the Navier-Stokes equation is considered. The equation is split in an Euler equation and a non-stationary Stokes equation within each time step. Unlike the conventional approach, we apply a non-homogeneous Stokes equation instead of homogeneous one. Under the hypothesis that the original problem possesses a smooth solution, the estimate of the H^s+1 norm, 0≦s<3/2, of the approximate solutions and the order of the L² norm of the errors is obtained.

In this paper a theorem of Main Boundedness Type is established and then used to study automatic continuity of derivations and module derivations on Banach algebras.

The present paper is the continuation of [1]. Some further generalizations of the fixed point theorems in [2] are obtained by means of the results in [1].

In [1] Homer introduced the honest polynomial reducibility and proved that under this new reducibility a set of minimal degree below O" is constructed under the assumption that P=NP. In this paper we will prove that under the same assumption a set of minimal degree can be constructed below any recursively enumerable degrees. So under the honest polynomial reducibility a set of low minimal degree does exist.

In this paper, the Dirichlet problem for the Monge-Ampére equation det(u_ij)= F(x,u,Δu) on a convex bounded domain Ω⊂R" is considered. The author establishes a new C^o-estimate and gives some new existence results. He also presents a new proof for the C^3,α-estimates of solutions, which not only weakens the smooth assumptions for F but also applies to more general nonlinear elliptic systems.

There are a few results of Welch (1967) and O. Moreno (1980) that count the number of solutions of Tr(y ^l)=0 in GF(2^m), for certain values of l. This paper counts the number of solutions of Tr(y^l)=h in GF(p^m), for further values of l. Then O. Moreno's question is answered.

Cowen-Douglas operators are important Fredholm operators of positive index, for which we can calculate the complete unitary invariants. On the other hand, the cyclicity of an operator is an important property, which provides a useful tool for studing this operator. This paper is devoted to prove: Every Cowen-Douglas operator is cyclic.

In this paper,we discuss the Chern characters of hypersurfaces with arbitrary singularities.When the hypersurface is smooth,the Chern characters are just the usual Chern numbers.For any given dimension,we prove that the Chern characters satisfy a kind of inequalities.And we discover that the Chern characters satisfy some kind of equalities when the dimension is greater than 3.Therefore we obtain some more inequalities which are satisfied by the Chern characters of hypersurfaces with arbitrary singularities.

The Grothendieck's criterion of weak compactness in space C(S) has been extended by C.P.Niculescu for weakly sequentially complete Banach lattices.This paper first provides some sufficient and necessary conditions for weakly sequentially complete Banach lattices,with the result that the criterion C.P.Niculescu obtained of the weak compactness has actually characterized the weakly sequentially Banach lattices.At the same time we extend C.P.Niculescu's important results.

Then we solve negatively Problem 1.11 which was posed by C.P.Niculescu,and a sufficient condition for this is given which generalizes Pelczyński's result.

H.Bass,E.H.Connell and D.Wright^[1] settled a case for the Jacobian Conjecture.In the present paper we will give another proof of the case.

Although many criteria for testing the planarity of a graph have been found since the beginning of the thirties,this paper presents a new criterion described by Boolean technique which is proved in an independent way without any use of the criteria obtained before.

The topological Markov chain or the subshift of finite type is a restriction of the shift on an invariant subset determined by a 0,1-matrix,which has some important applications in the theory of dynamical systems.

In this paper,the topological Markov chain has been discussed.First,we introduce a structure of the directed gragh on a 0,1-matrix,and then by using it as a tool,we give some equivalent conditions with respect to the relationship among topological entropy,chaos,the nonwandering set,the set of periodic points and the 0,1-matrix involved.

We prove in this paper that any flow on the 2-torus with no singular points and periodic orbits which is generated by a vector field V=(P,Q) satisfying V∈C¹ or V∈C⁰ and P≠0 is uniquely ergodic.Then we give an expression of the rotation number by using an invariant measure of a flow.

Let x_k be the state variable (solution) of a stochastic difference equation.This paper gives the laws of iterated logarithm for {x_k} and {x_kx_k^τ},which being strongly correlated,are neither stationary nor ergodic.The results obtained are then applied to Kalman filter and LQG control problem.

A conjecture of the author is verified by setting up a fundamental fixed point theorem so that some earlier results are unified and strengthened.

Let be a finitely generated extension field of Q,andα_i,β_j(1≤i≤m,1≤j≤n) be some complex numbers.Let (k=1,2,3) be fields obtained by adjoining to the numbers {α_i,β_j exp(α_iβ_j)},{α_i,exp(α_iβ_j)},and {exp(α_iβ_j)},respectively.In the present note the relation between the transcendental degree of over and the transcendence type of over Q is given.