We introduce the concept of weak Lyapunov similarity on linear systems. And we obtain that bounded linear systems with pure point spectrum is weak Lyapunov similar to constant coefficient diagonal linear systems.

Let G be a second countable locally compact Groupoid with Haar measure, H be a closed subgropoid of G, then we can get a Groupoid H\G2. We have proved C*(H) is Morita equivalent to C*(H\G2), which is a problem in [1]. Using the imprimitive bimodule and a homomorphism from C*(G) to M(G*(H\G2)) the induced representation from C*(H) to C*(G) is obtained.Particularly in the case of group bundles, we define a homomorphism from C*(H) to M(C*(G)) and obtain the integral form of the induced representation.

In this paper, we establish generalized inverses of functors. We obtain that: (1) generalized inverses of functors include generalized inverses of matrices by not looking matrices as morphisms,(2) generalized inverses of functors are different from generalized inverses of morphisms, but they have the same applications in many ways.

In this paper we gave some equivalent conditions of the fractional derivative and integral of holomorphic function f on Ω to belong to the mixed norm spaces H(p,q,)on bounded symmetric domains Ω of Cn, where 0 < p, q < and is a normal function. Applying these results, for two cases 0

Let R be a finite commutative local ring and let M denote the unique maximal ideal of R and k denote the residue field of R. In the present paper, the author determined the Sylow subgroups of the linear group GL.R, the symplectic group SP2nR and the hyperbolic orthogonal group O2nR. In general speaking, if chark = p, the Sylow p-subgroups of the three kinds of classical groups above are isomorphic to the subgroups generated by some special matrices respectively; if chark ≠ p, the Sylow p-subgroups of the three kinds of classical groups above are isomorphic to the wreath product of a cyclic group or a semidihedral group and finite terms of cyclic groups with type Zp.

In this paper, the compact-covering s-images of metric spaces and locally separable metric spaces are characterized. Michael-Nagami's question has been partially answered.

In this paper, we proved that the two conditions of linearly locally connectivity are invariant respectively under quasiconformal mappings which fixed the infinity and pointed out the condition-of fixed the infinity is essential by giving an example. On the other hand, we proved the converse of above results are true also.

It is very importent to study congruences and congruence extensions on semigroups. Many authors, such as Petrich, Pastijn, Jones, Trotter and Stralka etc.,put their most interests into this field. Some kinds of congruences on semigroups had been investigated by means of congruence pairs, see, for example, [1],[5],[6] and [7]. In particular, the descriptions for the maximum idempotent-separated congruences and the minimum group congruences on many kinds of semigroups were given. On the other hand, the class of semigroups with the congruence extension property was involved and described, see [4], [9] and [10]. In this paper, we introduce normalities and invariances for congruences on bands, and discuss that congruences on bands extend to fully regular subsemigroups of Hall semigroups of bands.In addition, we characterize congruences on bands, such as completely invariant congruences and characteristic invariant congruences on bands according to congruence extensions on semigroups, and describe the maximum congruence extension and the minimum congruence extension for a given congruence on a band.

In this paper we study the function f(x)= max B(u,u) - B(v, v)2, where M is a compact submanifold of a unit sphere, UM = ∪UMx is the unit tangent bundle on M, B is the second fundamental form of M. When M has parallel mean curvature,we obtain a Pinching theorem for the second fundamental form of M.If M is minimal immersed,we have the similar discussion.

In this paper isometries and hermitain deivations on nest subalgebras of factor Von Neumann algebras are discused. It is proved that every isometry is either an isomorphism multiply by a unitary operator or an anti-isomorphism multiply by a unitary operator. We give some conditions for which every hermitain derivation on nest subalgebras of factor Von Neumann algebras is inner, and show that every hermitain derivation on nest subalgebras of finite factor Von Neumann algebras is inner.

Let be the partially ordered set of all Turing degrees. A function j: D - D is said to be progressing if a ≤f(a) for all a ∈ D. For such a function f,we call a and b in D j-incomparable if a ≮ f(b) and b ≮ f(a), and f-comparable otherwise. The central result of the paper is that in ZFC the continuum Hypothesis 1 = 20 holds if and only if there exists a progressing function f: D → D such that any pair a and b in D is f-comparable.

A sufficient condition for the complete convergence of the maximum of U-statistics is discussed in this article.The author exactly elaboretes the links and difference between the complete convergence of the maximum of U-statistics and that of the commonly known sums of independent random variables, widering the range of the kernelfunctions and reducing the order of moments in Chow Yuanshih's and the author's another article.

Let R be a ring graded by any group G, H ≤ G. In [1], C. Nastasescu et al. discussed the smash product R#G/H with respect to finite G-set G/H. Our purpose in this paper is to consider a G/H-graded module and form a Morita context about R#G/H and R(H), with respect to any G-set G/H. We then apply them to discuss respectively the primitive, simplicity, prime and semiprime relating to R#G/H and R(H). In particular,we generalize some important results about group-graded rings in [2], [3] and [4].

By means of periodic transformations and integral manifold theory, we investigate planar periodic perturbed systems, and obtain for the strong resonant case , a condition under which an invariant torus is bifurcated from a weak focus of order one.

In this paper, we investigate the eigenfunction expansions for the generalized Sublaplacian.By introducing the generalized -twisted convolutions, we obtain the Plancherel theorem and a Hausdorff-Young inequality. Some results are given under the condition that the eigenvalues of the generalized Sublaplacian are discrete.

The purpose of this paper is to begin to lay the foundation of bicrossed products of Hopf algebra. The theory is motivated by asking that crossed product and crossed coproduct fit together to form a bialgebra. It is the further generalization of bismash product of Hopf algebra. We investigate the basic properties of these notions and give the exact sequence characterization of bicrossed product. We will see that any bicrossed product H with weakly inner action and inner coaction is isomorphic to a bicrossed product H A with trivial action ' and trivial coactionβ'.

In this paper, we deal with the long time behaviors of solutions for some quasilinear parabolic equations with nonlinear boundary conditions. The necessary and sufficient conditions on the global existence of solutions are obtained.

Associated with the endomorphism of the free semigroup S* generated by a finite alphabet S ={al, , at} there the collection of mappings on Rd and homomorphism f: S* Rd such that The generalized recurrent sets K [W] are constructed by means of those given above. Moreover the upper bound of the Hausdorff dimensions of K [W] are estimated when is the collection of homeomorphisms and the Hausdorff dimensions of K [W] are determined when is the collection of conformal contraction mappings.

In this paper the nonlinear approximation problem of set-valued maps in Banach space is investigated. The characterization theorems and uniqueness theorems are given for subsets, not only convex, of vector-valued continuous functions spaces.

In this paper the disconjugate existence theorem and the upper and lower solution existence theorem for the non-degenerative BVPs of the higher order singular ordinary differentialequations are established.

In this papers, we discuss the properties of medial idempotents on abundant semigroups, study quasi-adequate semigroups with a normal medial idempotent and some extreme cases of such semigroups, and give the description of structure of every type of such semigroups,respectively.

Consider the n order neutral differential equation where n≥1 is an odd integer. Sufficient conditions are obtained for the existence of a positive solution of Eq. (*) with p≠-1, also, sufficient and necessary conditions are obtained for the existence of a bounded positive solution of Eq. (*) with p =-1. These conditions are weaker for the function f. The results obtained answer partially an open problem in [1].

In this paper, Quasi-ZIF rings are defined. The structure of IF rings and Quasi-ZIF rings is studied. Furthermore, the relations between iF rings and Quasi-ZIF rings are discussed. Finally, IF rings are represented with Quasi-ZIF rings.

In this paper, we introduce the Sylow b-subpair graph to study the local fusion of Brauer subpairs. We prove the following theorems.Theorem 2 Let (D, bD) be a Sylow b-subpair of G. If F is a reduced family of b-Subpairs in (D, bD), then F is a conjugation family of G.Theorem 9 Let (D, bD) be a Sylow b-subpair of G. Assume that F is a family of b-subpairs which contains a full set of representatives {(P, bp)} of the conjugacy classes of b-subpairs with the following properties:(1) (P, bp) is a tame intersection b-subpair of G.(2) NG(P, bp)/PCG(P) contains a strongly embedded proper subgroup.(3) (P, bp) is a Sylow subpair of PCG(P).Then F is a conjugation family of G.The above results unify the Alperin-Broue and Puig's fusion theorems for blocks.

Weak convergence and strong consistency are established for a smooth version of the classical product-limit estimator of a distribution function when the data are subject to random censorship. The weak convergences oil D(I) = D(-∞,T] for T < TF=inf{x: F(x)=1} are shown to hold for the entire interval I under basically the same assumptions that have been used in the literature to establish the weak convergence of normalized estimator in D(I). Moreover,strong approximations and the laws of the iterated logarithm are proved.

In this paper, we discuss the nonpolynomial iterative equation G(fn0 (x), fn1 (x),…,fnk(x)) = F(x), the conditions are given for the existence, uniqueness of C1-solutions of the equations, and also there is a proved theorem for the continous dependence of C1-solutions of equation on the given function.

Let a, b be non-zero integers. Let p1,…,pr be distinct primes, and let P= {± |m1,…, mr are nonnegative integers}. Further let K be an algebraic number fieldof degree n with n ≥3, and let hk denote the class number of K. For α1,…, αm∈K with1 < m < n, let △(α1,… ,αm) denote the discriminant of α1,…,αm, and f(x1,… , xm) =μNK/Q(α1x1 +… + αmxm) ∈ Z[x1,…, xm]. In this paper we protve that if f(x1,…, xm) is non-degenerate and pi + △(α1,…,αm) for i=1,…, r, then the equation f(x1,…, xm) = by has at most (4sd2)2 (sd)6 (sd)6 integer solutions (x1,…,xm) satisfy gcd(x1,…,xm) = 1 andy∈P, where d= n!, s = r + ω(b) and ω(b) is the number of distinct prime factors of b.

In this paper, we prove that the isometrically algebraic isomorphism between two subalgebras of AF algebras with property of double commutant can be extended to an * isomorphism between their enveloping C* algebras. Consequently, the problem raised from [4] Remark2.12 can be solved affirmatively.

In this paper we give a simple proof of the following result:If a real-valued additive function f(n) satisfying ‖f(n + 1)- f(n)‖ = 0(1) (n→∞), where ‖x‖denotes the distance from x to the nearest integer, then there exists a contant C such that f(n) - C log n is a integral-valued function.

By the investigation of the zero-free region of Dirichlet L-functions and the meanvalue estimate of the distribution of primes, we proved in this paper that every odd integer N ≥ exp(exp(9.715)) can be written as a sum of three primes.

We first discuss the properties of right module D(D(X),I) of rings of differential operators on analytic irreducible curve X/k and its application. Then, we discuss the modules of finite dimensional vector spaces of rings of differential operators and holonomic modules on singular curves.

This paper proved that: If real Valued additive functions f(n) satisfies that ‖f(n+1)-f(n)‖=0(1) (n→co), where ‖‖denotes the distance of a real number to the nearest integer,then there exists a suitable constat c such that f(n)-clog n is an integer valned additive function. This confirms a conjecture of Katai Ⅰ.

In this paper we study the completeness of system of generalized eigenfunctions of A, which is the infinitesimal generator of a Riesz semigroup in a separable Banach space. In view of the resolvent and the spectral distribution of A, we present the criterion of completeness of system of generalized eigenfunction of A with resolvent of order p < +∞.

In this paper, we prove a result concerning a conjecture of Shorey and Tijdeman on the diophantine equation , and almost solve a conjecture of Edgar on the aboveequation.

In this paper we continue to study additive dervations of certain operator algebras.In section 2, we prove that every additive dervation of a nest-subalgebra of non-type In factor Von Neumann algebra is automatically continuous. This extends the main result of [1]. In section 3, we give a necessary and sufficient condition for which every additive dervation of two elements generated subspace lattice algebra is automatically linear.

There is on S6 an almost complex structure of canonical type. So, one can introduce the concept of Kahler angle. In the present paper, we study the minimal surfaces in S6 of constant Kahler angles. As the result, we obtain a rigidity theorem, from which one sees readilythat Simon's conjecture holds for these surfaces.

Let 0 ≤B≤ A for the Lowner's partial ordering of quateraion matrices. In this paper, we give a representation formula of the solution set B{1;≥ ;≥ a; A(1)} of a monotonicity.Some characterizations of three monotonicity conclusions of the generalized inverses of A and Bare obtained.

In this paper, under weaker condition, the authors prove the existence and uniqueness of solution of an operator equation which is positive α-homogeneous (α> 1) and give its application.

The type-changed operators are introduced in this paper. The order of a class of partial differential equations with multiple characteristics can be changed through the operators;therefor, such type-changed operators are powerful tools in studying some equations with multiple characteristics.As an example, the solution of the non-standard Cauchy problem (only one initial datum)for an equation of second order with multiple characteristics is gotten, and the uniqueness is proved.

In this paper, we characterise the completely distributive law in lattices with the Scott topology, the Zariski topology and the interval topology. We give some each-other-conditioned relations of existness of special elements and connectedness of the interval topology in a lattice.We also construct some important counterexamples.