In this paper, we will use the embedding flows in [1], [2] to give a complete description of the smooth centralizers and iterate radicals of all C^r (r≥2) Morse-Smale diffeomorphisms of the circle S¹. As a result, we prove that every centralizer is a solvable subgroup of Diff^r (S¹).

We estimate the kinematic measure of one convex domain moving to another under the group G of rigid motions in Rⁿ . We first estimate the kinematic formula for the total scalar curvature ∫_{∂D0∩g∂D1} Rdv of the n-2 dimensional intersection submanifold ∂D₀∩g∂D₁. Then we use Chern and Yen's kinematic fundamental formula and our integral inequality to obtain a sufficient condition for one convex domain to contain another in Rⁿ (≥4). For n=4, we directly obtain another sufficient condition in R⁴.

For a seemingly unrelated regression system with the assumption of normality, a necessary and sufficient condition for the existence of the Uniformly Minimum Risk Unbiased (UMRU) estimator of regression coefficients under strictly convex loss is obtained; it is proved that any unbiased estimator can not improve the least squares estimator; it is also shown that no UMRU estimator exists under missing observations.

In this paper three Banach spaces A₀(φ), A_∞(φ) and A¹(φ) of functions holomorphic in the unit ball B of Cⁿ are defined. We exhibit bounded projections from C₀(B) onto A₀(φ), from L¹(B) onto A¹(φ), and from L^∞(B) onto A_∞(φ). Using these projections, we show that A₀(φ)^*≌A¹(φ) and A¹(φ)^*≌A_∞(φ).

Using the calculus of variations in the large, especially computing the category of the symmetric configuration space of symmetric N-body-type problems, we prove the existence of infinitely many symmetric noncollision periodic solutions about the symmetric and nonautonomous N-body-type problems under the assumptions that the symmetric potentials satisfy the strong force condition of Gordon.

In this paper, we prove that for the genus 4 fibration, the relative canonical algebra is generated in degrees ≤3.

In this paper, the author studies the regularity of Munn rings and completely 0-simple semigroup rings. When either (i) R is a ring with identity and I and Λ are infinite, or (ii) R is a strong IBN and fully Dedekind-finite ring with identity and either I or Λ is finite, in Section 2, the regularity of the Munn ring M(R; I, Λ; P) is characterized; in Section 3, for a completely 0-simple semigroup S=M°(G; I, Λ; P), the regularity of RS is characterized. Meantime, the author shows that for a locally finite monoid S and a ring R with identity, RS is a strong IBN ring if and only if R is so; RS is fully Dedekind-finite if and only if R is so.

Let S(d, c) be the Dedekind sums. It is a classical problem to determine the distribution of points (d/c, S(d, c)) in the plane. In the present paper, we establish a quantitative theorem for the problem.

Let B_α(a) be an additive function on a ring of integers in the quadratic number field Q(√d) given by B_α(a)=Σ_p|a^*N^α(p) with a fixed α>0, where the asterisk means that the summation is over the non-associate prime divisors p of an integer a in Q(√d), N(a) is the norm of a. In this paper we obtain the asymptotic formula of Σ_N(a)≤x^*B_α(a) in the case where the class-number of Q(√d) is one.

In this paper, we generalize the Hall-Higman's reduction theorem by dropping the restrictive hypothesis (|G|, |H|)=1 and determine the detailed structure of G.

In this paper, an answer is given to a problem proposed by K.C. Li. This problem comes from the variable transformation for fitting a regression model.

The local existence of multiple shock fronts for hyperbolic conservation laws in higher dimensional space is established under the assumption that its frozen problem produces multiple uniformly stable planar shock fronts.

By rewriting the projection operator P₀ in wavelets in another formula, we obtain a characterization of dim J_V0 (x) where V₀ is a Γ-shift-invariant subspace of L²(Rⁿ) derived from a dual wavelet basis and prove that there does not exist a wavelet function ψ∈L²(R) such that ψ has compact support and ∪_k∈Z(suppψ+4πk)=R up to a zero subset of R.

This paper shows that W^1,p-quasiconvexity is a necessary and sufficient condition of swlsc (sw^*lsc) for multiple integrals with a Caratheodory function as the variational function,bounded by

whereg(x,u) is also a Caratheodory function,or

The result for W^k,p-quasiconvexity is also shown.

We consider perturbations which may or may not depend explicitly on time for the three-dimensional homoclinic systems.We obtain the existence and bifurcation theorems for transversal homoclinic points and homoclinic orbits,and illustrate our results with two examples.

Let λ₁,λ₂,...,λ_s bes non-zero real numbers not all of the same sign and not all in rational ratio,andk be a natural number; letD(k) be the leasts for which the inequality < (maxn_j)^-δ has infinitely many solutions.In this paper,we give some new estimations of D(k) for k≥6.

In this paper,we get the existence of a weak solution of the following inhomogeneous quasilinear elliptic equation with critical growth conditions:

whrer

Denote by E(X) the number of integers n≤X which are not a sum of a prime and a square.Set θ=1-0.994428b,where b≤min(0.9365/c₃,0.02578/c₄),c₃,c₄ being given in Lemma 8.The result E(x)<<x^θ is proved in this paper.

In this paper,we study the asymptotics of the spectrum of the Dirichlet (or Neumann) Laplacian in a bounded open set Ω ⊂Rⁿ (n≥1) with irregular but nonfractal boundary ∂Ω.We give a partial resolution of the Weyl conjecture,i.e.for the counting function N_i(λ)(i=0:Dirichlet;i=1:bNeumann),we have got a precise estimate of the remainder term ψ_i(λ)=ψ(λ)-N_i(λ) for largeλ,whereψ(λ) is the Weyl term.This implies that for the irregular but nonfractal drum Ω,not only the volume |Ω|_n is spectral invariant but also the area of boundary |∂Ω|_n-1 might be spectral invariant as well.

The purpose of this paper is to study the long time asymptotic behavior for a nonlinear Schrödinger equations with magnetic effect.Under certain conditions,we prove the existence and nonexistence of the non-trivial free asymptotic solutions.In addition,the decay estimates of the solutions are also obtained.

In this paper,we have constructed a random weighting statistic to approximate the distribution of studentized least square estimator in a linear regression model with ideal accuracy o(n^-1/2).Thus,we have provided a more practical distribution approximating method.

In this paper a kind of theta function is constructed by means of spherical function.And we also obtain some Hilbert modular forms of half integral weight.

This paper introduces three kinds of operators on planar graphs with binary weights on edges,for which combinatorial invariants on two kinds of equivalences are found.Further,it is shown that the Jones polynomial and the bracket polynomial which are proved to be new topological invariants on knots in topology become special cases.Moreover,these invariants are a kind of generalization of Tutte polynomial on graphs.

Graham Higman posed the question: How small can the integers p and q be made,while maintaining the property that all but finitly many alternating and symmetric groups are factor groups of Δ(2,p,q)=<x,y:x²=y^p=3 (xy)^q=1>? He proved that for a sufficiently large n,the alternating group is a homomorphic image of the triangle group Δ(2,p,q) where p=3 and q=7.Later,his result was generalized by proving the result for p=3 and q≥7.Choosing p=4 and q≥17 in this paper we have answered the "Hiqman Question".

In this paper,we show that,without involving the C^*-bundle theory,elementary differential topology can do the classification of homogeneous C^*-crossed product C(X) x_α Z,where X is a compact differentiable manifold of low dimension and α is a diffeomorphism of X.The motivation of this work is the Shultz's theorem which states that a C^*-algebra can be identified with its pure state space carrying w*-topology and certain geometric structure.

Let X be a generic smooth irreducible complex projective curve of genus g with g≥4.In this paper,we generalize the existence theorem of special divisors to high dimensional indecomposable vector bundles.We give a necessary and sufficient condition on the existence of n-dimensional indecomposable vector bundles E on X with det (E)=d,dim H ⁰(X,E)≥h.We also determine under what condition the set of all such vector bundles will be finite and how many elements it contains.

In this paper,we obtain some formulas for determining the Hausdorff dimension of the inverse image of compact sets under general Levy processes in Rⁿ.

Let M⊆Rⁿ⁺¹ be a compact connected smooth hypersurface,and let W⊆Rⁿ⁺¹ be the area bounded by M.We study the question:Does W contain a principal centre of curvature for some point of M?

According to the Projection Pursuit (PP) method and the random weighting method,we propose a PP random weighting method,and set up the asymptotic distribution theory and strong limit theorem of PP random weighting empirical process.Applying this method,we obtain two kinds of goodness-of-fit test for a multivariate distribution function,i.e.,we get the random weighting approximations of PP Kolmogorov Smirnov statistics (PPKS) and PP Smirnov Cramér Von Mises statistics (PPSC),we prove that the asymptotic distribution of PPKS and PPSC are the same as those of their respective random weighting approximations.

Under a generic assumption,the existence and the uniqueness of the periodic orbit generating from a homoclinic bifurcation are shown,and the dimensions of its stable and unstable manifolds are given.In the case of a 3-dimensional system,our result revises the stability criterion given in [4,5].

For the N-parameter d-dimensional Generalized Brownian Sheet (in short G.B.S.) satisfying some conditions,this paper proves that when 2N≤αd,with probability one,∀E ∈ß(R₊^N ),there holds

.Especially when α=β=1,we have dim ?(E)=2 dim E.Noting that even if α=β=1,the G.B.S.is wider than the Brownian Sheet,thus we have extended the uniform dimension result of Mountford,T,S.^[1].

In this paper,for any given natural numbersn anda,we can construct explicitly positive definite indecomposable integral Hermitian forms of rank n over Q with discriminant a,with the following ten exceptions:n=2,a=1,2,4,10;n=3,a=1,2,5;n=4,a=1;n=5,a=1;andn=7,a=1.In the exceptional cases there are no forms with the desired properties.The method given here can be applied to solving the problem of constructing indecomposable positive definite Hermitian R_m-lattices of any given rank n and discriminant a,where R_m is the ring of algebraic integers in an imaginary quadratic field Q with class number unity.

Let G be a finite group.If for every primer,where R₁∈Syl_rG and R₂∈Syl_r (L_n(q)),then G≌L_n(q).

We first give an example to illustrate that the results in [12] concerning the boundedness of solutions of nonlinear oscillatory equations are not true.And then we obtain sufficient or necessary conditions for the boundedness of solutions of the nonlinear system of differential equations

without the usual signum conditions xg(x)>0 for all x≠ and f(x)≥0 for all x∈R.

In this paper the global behavior of a class of three-dimensional competitive systems is investigated.The existence of semi-stable periodic orbits is proved.A generalization of the results of [1-4] is obtained.

As one of main backgrounds of locale theory,topologies have close connections with locales.But locales have other backgrounds such as algebra,mathematical logic,etc.So there are many differences between locales and topologies.Spatiality is an important localic property to investigate the connections between locales and topologies.The T_D property is a special separation property which plays an important role in this kind of investigations.Just as it will be proved in this paper,the T_D property often appears as the lowest requirement for many topological spaces such that they can be described with localic properties and vice versa.In this paper,we show these special properties of the T_D axiom and investigate some other interesting and important problems of T_D-spatiality of locales.

In this paper,we mainly prove that everyn-dimensional Lip microbundle over a locally finite simplicial complex is micro-identical to a Lip-Sⁿ(Rⁿ)-bundle,and any two such are micro-identical,isomorphic Sⁿ(Rⁿ)-bundles.

This paper is devoted to study Cauchy problems of multidimensional semilinear strictly hyperbolic equations of second order with strongly singular initial data,where the derivatives of the initial data have discontinuity on two smooth curves transversally intersecting each other.The existence of the solution is proved,meanwhile,it is precisely discribed the flowery structure of the singularity of the solution.

This paper includes two parts.In the first part,we deal with the relations between the orders,the lower orders and the number of Julia directions of entire functions.In the second part,we deal with the existence of the zeros of a class of meromorphic functions.This class of meromorphic functions has an interesting physical background.

In this paper,we explicitly describe all the inverses and pseudo-inverses of a strong endomorphism of a graph.The number of them is determined.In addition,we give a characterization of a strong endomorphism whose pseudo-inverse set coincides with its inverse set.The graph,each strong endomorphism of which has this property,is also investigated.