The so-called "principal term" of Goldbach problem, i.e., the principal order of the number of representations of an even number as a sum of two primes, is evaluated here in a way quite different from the classical circle method. It is amazing that the author did this with the knowledge just at the PNT (Prime Number Theorem) level, even without using high-level theorems in analytic number theory as usual, e.g., Siegel-Walfisz Theorem.

Given a Markov process satisfying certain general type conditions, whose paths are not assumed to be continuous. Let D be an open subset of the state space E. Any bounded function defined on the complement of D extends to be a function on E such that it is harmonic in D and satisfies the Dirichlet boundary condition at any regular boundary point of D. The relation between harmonic functions and the characteristic operator of the given process is discussed.

This paper is concerned with Smale diffeomorphisms on compact oriented surfaces. We determine with which genus a closed oriented surface can carry a Smale diffeomorphism associated with a subshift of finite type corresponding to a given reciprocal polynomial with constant term one, and determine what a combination of subshifts and period imformations of sinks and sources can be realized on a given oriented surface by a Smale diffeomorphism in a definite isotopic class. Necessary and sufficient conditions are given.

The main result in this paper is the following:
Theorem. Assume that W is a k-connected compact PL n-manifold with boundary, BdW is (k-1)-connected, k≥1(BdW is 1-connected for k=1), 0≤ h≤2k, 2 n-h>5 and there exists a normal block (n-h-1)- bundle v over W, then(1) There is a neat PL embedding W→ D{D^2n-h} which normal block bundle is isomorphic to v⊕ε.(2) There is a PL embedding W→S^2n-h which normal block bundle is isomorphic to v.
Where ε denotes the trivial block 1-boundle, D^2n-h={x=(x₁, x₂,…, x_2n-h)∈ R^2n-h||x_i|≤1} and S^2n-h-1=BdD^2n-h.

The well-known Bombieri-A. I. Vinogradov theorem states that
(1)
where A is an arbitrary positive constant, B=B(A)>0, and as usual,

Λ being the Von Mangoldt's function.

Let f(x)∈ L_p[0,1], 1≤ p≤∞. We shall say that function f(x)∈Δ^k (integer k-1) if for any h∈[0, 1/k] and x∈[0,1-kh], we have Δ_h^k f(x)≥0. Denote by ∏ _n the space of algebraic polynomials of degree not exceeding n and define

We prove that for any positive integer k, if f(x)∈Δ ^k ∩L_p[0, 1], 1≤p≤∞, then we have

where C is a constant only depending on k.

In this paper we get the invariant functions and the invariant harmonic functions under Aut (D) for certain Reinhardt domains D. By using the invariant functions, we get much more invariant Kahler metrics. And the Ricci curvatures, scalar curvatures and holomorpic sectional curvatures are also obtained, which are very different for the bounded homogeneous domains.

The purpose of this paper which is a sequel of "Boolean planarity characterization of graphs" ^[9] is to show the following results.(1)Both of the problems of testing the planarity of graphs and embedding a planar graph into the plane are equivalent to finding a spanning tree in another graph whose order and size are bounded by a linear function of the order and the size of the original graph, respectively.(2)The number of topologically non-equivalent planar embeddings of a Hamiltonian planar graph G is
τ(G)=2^c(H)-1,
where c (H) is the number of the components of the graph H which is related to G.

Let D' be a positive integer which is square free, and let h' (-D') denote the class number of the imaginary quadratic field 2e (√-D'). In this paper we prove that if D'> exp exp exp 1000, D'=4k″-a², where a, k and n are integers with a>0, k>1, n>1, then

except possibly for n=3, D'=3(2 a₁+1)²±4, k=(2 a₁+1)²±1, a=2 a₁+1, where a₁ is an integer.

In this paper we deal with the measure model with independently distributed errors. For estimating the error distribution of the sample mean, random weighting method is utilized. It is proved that the accuracy of the approximation is of the order of o (1/√n), which is the ideal rate.

In this paper we have proved that the Lie algebra L over a field F of characteristic 0 which admits a fixed-point-free automorphism of finite order is solvable, and L is nilpotent if L admits a fixed-point-free automorphism of prime order. The problem seems to be open if the field F has prime characteristic.

In this note we improve a recent result of Nualart and Zakai [1] on the existence of density of the law of a Wiener functional.

In this paper,by means of Morse theory of isolated critical points (orbits) we study further the critical points theory of asymptotically quadratic functionals and give some results concerning the existence of multiple critical points (orbits) which generalize a series of previous results due to Amann,Conley,Zehnder and K.C.Chang.As applications,the existence of multiple periodic solutions for asymptotically linear Hamiltonian systems is investigated.And our results generalize some recent ones due to Coti-Zelati,J.Q.Liu,S.Li,etc.

A simple qualitative model of dynamic combustion
(1)
is considered,where constant q>0 represents the binding energy.The existence and uniqueness of classical solution for the initial value problem and for the first boundary value problem of system (1) are proved and then it is proved that as ε→0 classical solutions of the initial value problem of system (1) converge to a pair of limit functions,which is an admissible solution of the initial value problem of system (1) with ε=0.

For a variety X in Pⁿ,we define a number b₂,called coordinate gap number,and prove that b₂>2 only if X is not reflexive.Then,for a smooth surface in P³,we obtain a concrete sufficient and necessary condition for b₂>2,which enables us to discuss the biduality of surfaces in P³.

Let A and R be commutative rings,and m and n be integers≥3.It is proved that,if Λ:St_m(A)→St_n(R) is an isomorphism,then m=n.When n≥4,we have: (1) Every isomorphism Λ:St_n(A)→St_n(R) induces an isomorphism λ:E_n(A)→E_n(R),and Λ is uniquely determined by if St λ; (2) If St_n(A)≡St_n(R) then K_2.n(A)≡K_2.n(R); (3) Every isomorphism E_n(A)→E_n(R) can be lifted to an isomorphismSt_n(A)→St_n(R); (4)St_n(A)≡St_n(R) if and only if A≡R.For the case n=3,if St₃(A) and St₃(R) are respectively central extensions of E₃(A) and E₃(R),then the above (1) and (2) hold.

This paper deals with the implications of the boundedness of a solution for the periodic dynamical system of second order.

A discrete group can generate a C^*-algebra,denoted by C_β^*(G),by considering the so called conjugation regular representation β of the group G.Based on this treatment,the inner amenability of G shall be characterized by the existence of the states on C_β^* (G) satisfying some certain conditions.

In this paper we discuss the bifurcation of homoclinics of the equation
(*)
whereg (x) is such that the unperturbed equation x"+g (x)=0 has homoclinic orbits through zero.We give the bifurcation graph of small parameters μ and λ,and that of small functions g₁ and f₁.Then we give a criterion to determine the codimensions of bifurcation manifolds of small functions g₁ and f₁.Thus we generalize the conclusions of J.K.Hale.

Consider the weighted sums of a sequence {X_n} of independent random variables or random elements inD [0,1].For convergence of S_n in probability and with probability one,in [2],[3] etc.,the following stronger condition is required: {X_n} is uniformly bounded by a random variable X,i.e.P(|X_n|≥x)≤P(|X|≥x) for allx>0.Our paper aims at trying to drop this restriction.

A geometrical index for group Z^p as well as its applications is given,where p is a prime integer.

In this paper,by means of our Z_p index theory developed recently we study the existence of multiple periodic solutions for asymptotically linear nonautonomous wave equations.All previous known results rely either on the oddness of nonlinear terms,or on autonomous systems,and the best result for the general case is the existence of two nontrivial periodic solutions (Amann,Zehnder,K.C.Chang,S.P.Wu,S.J.Li,Z.Q.Wang).In this paper,under the assumption that the nonlinear term is T/p periodic we discuss multiple periodic solutions of nonautonomous systems and generalize a series of previous results.

In this paper,we discuss n-dimensional Lotka-Volterra prey-predator chain systems and prove that the locally,asymptotically stable equilibrium point of a five-dimensional system is a globally stable one.Namely,we prove that Chen Lansun's conjecture is true for the case n=5.

Let X_i (i=1,2,...) be the independent random variables on the probability space (Ω,π,P).In this paper we will show that the necessary and sufficient condition of the uniform convergence of Σ X_i is the convergence of Σ m_i⁽¹⁾ and Σ m_i⁽⁰⁾,where m_i⁽¹⁾,m_i⁽⁰⁾ denote the 1-quantile and 0-quantile of X_i respectively.

For H∈C² (Ω,R) where 0∈ΩR²ⁿ,Hⁿ (0)=0 and det H"(0)≠0,the paper proves that there is a global Hopf bifurcation from x=0 for Hamiltonian system x=JH(x) iff JH(0) possesses purely imaginary eigenvalues.The work improves the corresponding result of J.C.Alexander and J.Yorke (Amer.J.Math.,100 (1978),263-292).

This paper is concerned with the Douglas problem in general Riemannian manifolds.For its perturbed problem we establish a uniform a priori estimate under the energy level min {d^*,m^*+s₀} (see Theorem 3.3).This level seems to be the best possible since in case of Rⁿ it is just the Douglas condition.

Let M be a C²-Finsler manifold modeled on a Banach space,and let f be a C²-real-valued function defined on M.Using the A-gradient vector field which was introduced in [31] we give a suitable definition for nondegenegacy of critical points of f,then generalize the Morse handle-body decomposition theorem and the Morse inequalities to a kind of Banach manifolds.A generalization in the reflexive case has been done in [31].

In this paper we treat a variation of the (B,N)-Pair Theorem.We investigate the amalgam (S₅,Chev (3)),whose structure is similar to that of a Weak (B,N)-Pair,and thus give a description of the 3-local subgroups of M^c-L and U₅ (2).

In this paper,we discuss the existence and uniqueness of solution of an equation f(x)=a in a Banach algebra.We have the following fundamental lemma:

Let A be a Banach algebra with identity,U be an open subset of complex plane C,f be an univalent function in U,V=f(U),a∈A.If the spectral set of a,σ (a)⊂V,then there exists an unique x∈A,such that
σ(x)⊂U,and f (x)=a.
Using this lemma,we can get generalizations of corresponding results in [1-8].Moreover,we also give a generalization of the fundamental theorem on special solution in [3].

Let p be an odd prime with p≡1 (mod 4) and Q(√p) be the real quadratic field.Also let ε and h denote the fundamental unit and the class number of Q(√p),respectively.The main purpose of this paper is to study the "explicit" expressions of ε^h and ε^2h,and to discuss the problems related to the conjecture of Ankeny-Artin-Chowla.

In the paper, we give the almost sure limiting behaviour of the increments of partial sums of a ψ-mixing sequence when the moment generating functions do not exist. The results obtained are close to those for an independent sequence provided that the mixing coefficients tend to zero rapidly enough.

Let A be a finitely generated commutative Z-algebra with Krull dimension d, and let π be an arbitrary finite group. It is proved that the Steinberg group St_n (Aπ) is finitely presented whenever n≥4. If, in addition, n≥d+3, and K₁ (Aπ) and K₂ (Aπ) are finitely generated, then E_n (Aπ) and GL_n (Aπ) are finitely presented.

In this paper, we determine the groups (k_i are odd), (k_i are odd and (k_i/2)=0),J_n^{2,k₁,…k_f} (k_i are even and n>k_l), (k_i are even and n>k_l), (k_i are even and n>k_l, k_l≥12), J_n^1,2, J_n^2,3, J_n^1,4. And we obtain the relation Im σ_n^k =J_n^l,k.

In the present paper a few coincidence theorems and minimax theorems are given so as to unify and strengthen the corresponding results in [3], [4] and [5].

This paper investigates the enumerative problems of flagged skew plane partitions in which each row (column) has an upper and a lower bounds on the entries. By means of dominance technique, a direct and elementary derivation for their generating functions is presented which may be more accessible to readers.

In the present paper a form of generalization of Gelfond's lemma on dense sequences of polynomials is proposed. For a set of complex numbers θ₁, …, θ_s we define the coefficients_gk(θ₁, …, θ_s) (0≤k≤s) and give the relations between them and the transcendental degrees or the transcendence types of the field O (θ₁, …, θ_s) or its subfields.

The authors prove in this paper the following.
Theorem.Let f (Z) be any transcendental meromorphic function in the plane. Then for any given non-constant fractional linear function ψ (Z), the set

is at most a countable set.

In this paper we prove that an affine hypersphere with scalar curvature zero in a unimodular affine space of dimension n+1 must be contained either in an elliptic paraboloid or in an affine image of the hypersurface x₁x₂…x_n+1=const. We prove also that an affine complete, affine maximal surface is an elliptic paraboloid if its affine normals omit 4 or more directions in general position.

In this paper,we give a modified proof of Sullivan's eventual theorem for rational dynamics. Our proof is based on some idea of the Sullivan's proof,but does not make use of the Teichmuller theory.

In this paper we get an existence theorem of nontrivial critical points by using the local linking idea. As applications, we study the existence of nontrivial periodic solution of Hamiltonian systems.