In this paper, we have discussed the one sided topological Markov chain and proved the following conditions to be equivalent: 1. topologically strongly mixing, 2. topologically weakly mixing, 3. topological transitivity and the existence of two periods which are co-prime. As a consequence, we have come to the conclusion that mixing implies positive entropy but the converse is not true.

  We discuss the reflection of weak singularities of solutions to semilinear hyperbolic systems by using the method of energy estimates in the space of conormal distributions, and consequently obtain the general results on the reflection of weak singularities of the solution to a high order semilinear strictly hyperbolic equation.

 By means of the Durfee rectangles of partitions, a very elementary proof for Jacobi's triple product identity and its finite analogue is presented.

 This paper is devoted to the combined Fourier spectral and finite element approximations of three-dimensional, semi-periodic, unsteady Navier-Stokes equations. Fourier spectral method and finite element method are employed in the periodic and non-periodic directions respectively. A class of fully discrete schemes are constructed with artificial compression. Strict error estimations are proved. The analysis shows also that the classical two-dimensional velocity-pressure elements can be readily extended to solving such three-dimensional semi-periodic problems, provided they satisfy the two-dimensional “inf-suf” condition.

 LetV _n (Fq) be the orthogonal geometry of dimensionn overFq, whereq is a power of 2 andq>2. LetA be the arrangement inV _n (Fq) consisting of all (n−1)-dimensional non-singular subspaces ofVn(Fq) and letL(A) be the geometric lattice consisting of all intersections of elements ofA. In this paper we give a characterization of the elements ofL(A) and compute the characteristic polynomial ofL(A).

Coset diagrams for the orbit of the groupG=?x,y∶x ²=y ⁴ =1? acting on real quadratic fields give some interesting information. By using these coset diagrams, we show that in the orbitpG, where , the non-square positive integern does not change its value and the real quadratic irrational numbers of the formp, wherep and its algebraic conjugate have different signs, are finite in number and that part of the coset diagram containing such numbers forms a single closed path, which is the only closed path in the orbit ofp.

The distortion problem ofK-quasiconformal mappings of the unit diskD={z∶|z|<1} onto itself with the original fixed is discussed, a new estimate is given.

 In this paper, we study harmonic maps into ellipsoids and generalize some interesting results on harmonic maps into spheres of R. Schoen and K. Uhlenbeck.

 In this paper, we prove that_χ(Seq_ξ)=d, when ξ is Frechet filter orP-point in ω^* withx(ξ,ω^*)≤d.

 In this paper, a multiple solution theorem for minimal annuli coboundaries in a Riemannian manifoldN is established. Especilly, when the target manifoldN is the standard sphereS ⁿ , it implies the existence of at least two minimal annuli with given pair of wires (Γ¹, Γ²) as their common boundaries θ.

 In this paper we shall study the complete Dirichlet character sums involved with some polynomials and rational functions whichare useful to the Waring's problem.

 The CUSUM rule and Shiryayev-Roberts (S-R) rule, proposed by Page and Shiryayev (or Roberts) respectively, are two asymptotically optimal rules of detecting a change in distributions. This paper is concerned with the properties of their moments. In the context of continuous times, explicit formulas of their high moments are established.

 In this paper, we obtained the sufficient and necessary condition for the unique existence of periodic solution of the linear Volterra integro-differential equations of the form

. We also proved that the mentioned equation has unique periodic solution is a generic property.

 Suppose that

is a Clifford matrix of dimensionn, g(x)=(ax+b)(cx+d) ⁻¹. We study the invariant balls and the more careful classifications of the loxodromic and parabolic elements inM(R ⁿ ), prove that the loxodromic elements inM(R ^2k+1 ) certainly have an invariant ball, expound the geometric meaning of Ahlfors' hyperbolic elements, and introduce the uniformly hyperbolic and parabolic elements and give their identifications. We prove that

These results are fundamental in the higher dimensional Möbius groups, especially in Fuchs groups.

 In this paper, we use the variational method to study the existence of periodic and generalized periodic solutions of planar second order Hamiltonian systems when a singular potential is present. The results obtained are applied to the study of generalized periodic solutions of the Restricted Three-Body Problem and the Planarn-Body Problem.

 From a coincidence theorem in [1] there are derived some interesting applications to many respects such as minimax inequality, fixed point theory, existence of minimizable quasiconvex functions, etc..

  In this article, we prove that the repetitive algebra of an iterated tilted algebra can be realized by the repetitive algebra of some tilted algebra. In particular, an iterated tilted algebra can be obtained by a series of reflections from some tilted algebra.

 In this paper, we consider an initial value problem for nonlinear integro-differential equations in a Banach space. First, we give a comparison result between the under and over functions and some comparison principles. Then, using these results and the Kuratowski measure of noncompactness, we establish the existence theorem of extremal solutions between the under and over functions, and prove that there exists a unique solution between the lower and upper solutions under an additional Lipschitz's condition.

It is a general problem to study the measure of Julia sets. There are a lot of results for rational and entire functions. In this note, we describe the measure of Julia set for some holomorphic self-maps onC ^*. We'll prove thatJ(f) has positive area, wheref:C ^*→C ^*,f(z)=z ^m c ^P(z)+Q(1/z) ,P(z) andQ(z) are monic polynomials of degreed, andm is an integer.

 The main purpose of this paper is to study the existence of nonoscillatory solutions of the second order non-linear differential equation (1). The author first generalizes a Wintner's lemma [1,8] to nonlinear equations (i.e. the following Theorem 1 and 4), and then obtains the necessary and sufficient conditions for the existence of nonoscillatory solutions of (1). These theorems generalize the corresponding results of [1] to include nonlinear equations. Using the above results, the author further obtains a series of criterion theorems for the existence of nonoscillatory solutions and comparison theorems for the oscillation and nonoscillation of nonlinear equations.

 ConsiderD _n the maximum Kolmogorov distance betweenP _n andP among all possible one-dimensional projections, whereP _n is an empirical measure based ond-dimensional i.i.d vectors with spherically symmetric probability measureP. We show in this paper that

for large λ,d≥2 and an appropriate constantc ₁. From this, when dimensiond is fixed, we give a negative answer to Huber's conjecture,P(D _n >ε)≤N exp(−2n? ²), whereN is a constant depending only on dimensiond.

In this paper, we give an explicit expression of the fundamental solutions and the global solvability for a class of LPDO's consisting of left invariant vector fields on the nilpotent Lie groupG d ₁+d ₂.

  In this paper some necessary and sufficient conditions are obtained for a operator tensor product to be zero, compact, normal, hyponormal, subnormal, essential normal,k-quasihyponormal, etc.

 This paper discusses the Keldys-Fichera boundary value problem for a kind of degenerate quasilinear elliptic equations in divergence form. The existence theorem, comparison principle and uniqueness theorem are proved.

In this paper Ruelle's inequality for the entropy of diffusion processes and, more generally, for the entropy of random diffeomorphisms is proved.

  One new notation, called theA-prompt permitting, is defined in the Turing reducibility, and its basic properties are given, by which we shall show that two r.e. setsA andB are one minimal pairs if and only ifB is notA-prompt permitting. It is proved that there are no maximal minimal pairs.

We will solve several fundamental problems of Möbius groupsM(R ⁿ) which have been matters of interest such as the conjugate classification, the establishment of a standard form without finding the fixed points and a simple discrimination method.

Let be a Clifford matrix of dimensionn, c ≠ 0. We give a complete conjugate classification and prove the following necessary and sufficient conditions:g is f.p.f. (fixed points free) iff , |α|<1 and |E−AE ¹| ≠ 0;g is elliptic iff , |α| <1 and |E−AE ¹|=0;g is parabolic iff , |α|=1; andg is loxodromic iff , |α| >1 or rank (E−AE ¹) ≠ rank (E−AE ¹,ac ⁻¹+c ⁻¹ d), where α is represented by the solutions of certain linear algebraic equations and satisfies

 In this paper, we prove the regularity near the origin of the ν-function of Dirac operators associated with certain non-torsion free connections.

 It is proved in this paper that there exists an incomplete Mendelsohn triple system IMTS(u,v; λ) if and only ifλ(u-v)(u-2v-1)≡0(mod 3),u≥2v+1 and (u, v, λ) ≠ (6, 1, 1). As a consequence, it is proved that for any given λ≥1, a Mendelsohn triple system MTS (v, λ) can be embedded in an MTS (u, λ) if and only ifλu(u-1)≡0(mod 3) andu≥2v+1.

 The purpose of this paper is to establish some basic results about perturbations and approximation of integrated semigroups.

 In this note, we'll establish the following Main Theorem. LetD be a locally finite intersection of bounded strictly pseudoconvex domains in a complex space. Then there exists a proper holomorphic mappingf fromD into some unit ball.

The fundamental theory of Generalized Itô's Stochastic Functional Differential Equations (GISFDE), namely, stochastic functional differential equations associated with a stochastic integral based onC-semimartingales, is discussed. The main purpose is to establish the concepts of GISFDE, weak solution, strong solution, solution-measure and several kinds of the uniqueness of solution, to show the relationships smong these solutions and get existence and uniqueness theorems of the solution.

 Necessary and sufficient conditions are given for oscillations of second order linear differential equationx″+p(t)x′+q(t)x=0.

 In this paper, we discuss the representation-finite selfinjective artin algebras of classB _n andC _n and obtain the following main results:

 Throughout this paperR will denote a ring with idenity element andM a unitary right module overR. AnR-moduleM is said to be direct injective if and only if given direct summandN ofM with injectioni _N:N→M and a monomorphismg:N→M, there exists an endomorphismf ofR-moduleM such thatfg=i _N.

  In this paper, we discuss the existence and nonexistence of solutions for the problem311-2 where Ω is a bounded smoothness domain inR ^N, γ ε R¹, μ>-0,f(x) is a given non-negative function. Some interesting resultus have been obtained.

 LetP(x) denote the greatest prime factor of II_z<n≤x+x1/2 n. In this paper, we prove thatP(x)>x ^0.723 holds true for a sufficiently largex.

 In this paper, under some growth condition on the volume of a geodesic ball with radiusr, we compute the essential spectrum of some class of Riemannian manifolds with finite volume. This result refines some earlier results of R. Brooks.

 LetG be a group andA aG-graded ring. A (graded) idealI ofA is (graded) essential ifI⊃J≠0 wheneverJ is a nonzero (graded) ideal ofA. In this paper we study the relationship between graded essential ideals ofA, essential ideals of the identity componentA _e and essential ideals of the smash productA#G ^*. We apply our results to prime essential rings, irredundant subdirect sums and essentially nilpotent rings.

 In the present paper there are given some existence theorems on simultaneous solutions to fixed point and minimax-type inequality problems, hence to fixed point and variational-type inequality problems.