ζ_k is one of the most important constant in the sieve methods. This paper gives the relatively accurate lower bound and upper bound on it, that is, , where c=1.22…and k>16.

In the beginning of 1980's Cohen, R. proved that h₀b(1k)=B(1,k)=B(1,k) ζ₁ survives to E_∞ in the Adams spectral sequence. Later, Cohen, R. and Goerss, P. proved that h₀h_j=ζ₁^pjζ₁ is a permanent cycle. And they are represented by ζ_k ,η_j respectively. Here the author proved:
Theorem 2: Let 2≤s≤p-1,j≥3, then β_s η_j ≠0.
Theorem 3: For 2≤s≤p-1,k≥2, β_s ζ_k ≠0 in the stable homotopy groups of spheres.

In this paper, the space D^p(Ω) of functions holomorphic on bounded symmetric domain of Cⁿ is defined. We prove that H^p(Ω)⊂D^p(Ω) if 0<p≤2, and D^p(Ω)⊂H^p(Ω) if p≥2, and both the inclusions are proper. Further, we find that some theorems on H^p(Ω) can be extended to a wider class D^p(Ω) for 0<p≤2.

Existence and uniqueness conditions for nonnegative solutions to initial value problems of general sublinear-linear differential equations are obtained. They extend the uniqueness theorem due to H.Murakami^[6] and the main results of H.G. Kaper and M.K.Kwong^[4].

It has been proved by the author that every p-generic degree is nonbranching. In this paper we construct an r.e. degree which is strongly nonbranching and non-p-generic. It means that the class of nonbranching degrees does not coincide with the class of p-generic degrees.

In this paper, we consider the related weighted boundedness of the operators which were studied by J. Duoandikotxea and J. L. Rubio de Francia.

The Multiplier Theorem is a celebrated theorem in the Design theory. The condition p>λ is crucial to all known proofs of the multiplier theorem. However in all known examples of difference sets μ_p . is a multiplier for every prime p with (p, v)=1 and p|n. Thus there is the multiplier conjecture: "The multiplier theorem holds without the assumption that p>λ". The general form of the multiplier theorem may be viewed as an attempt to partially resolve the multiplier conjecture, where the assumption "p>λ" is replaced by "n₁>λ". Since then Newman (1963), Turyn (1964), and McFarland (1970) attempted to partially resolve the multiplier conjecture (see [7], [8], [9]). This paper will prove the following result using the representation theory of finite groups and the algebraic number theory: LetG be an abelian group of order v,v₀ be the exponent of G, and D be a (v, k, λ)-difference set inG. If n=2n₁, then the general form of the multiplier theorem holds without the assumption that n₁>λ in any of the following cases:
(1)2|n₁;
(2)2 n₁ and (v, 7)=1;
(3)2 n₁,v, and t≡1 or 2 or 4 (mod 7).

The usual concept of cobordism is concerned with closed manifolds. In this paper, we generalise the concept of cobordism to the extent of bounded manifolds and obtain a homotopy theorem similar to the theorem of Thom.

We establish the existence of continous solutions of the first boundary value problem for nonlinear diffusion equations of the form under the conditions that is strictly increasing and m>n-1, where n is the space dimension. Moreover, the uniqueness is proved for solutions with some regularities.

In this paper we investigate the self-adjointness for a kind of pseudodifferential operators, which include the nonsemi-bounded Schrödinger operator, -Δ+v(x),v(x)→-∞, as │x│→∞, and the relativistic corrections to it, .

By discussing the zeros of periodic solutions we give in this paper a criterion for the existence of exactlyn+1 simple 4-periodic solutions of the differential delay equation X(T) = -f(x(t-1)).

In this paper, we consider the periodic solution problems for the systems with unbounded delay, and the existence, uniqueness and stability of the periodic solution are dealt with unitedly. First we establish the suitable delay-differential inequality, then study seperately the problems of periodic solution for the systems with bounded delay, with unbounded delay and the Volterra integral-differential systems with infinite delay by using the character of matrix measure and the asymptotic fixed point theorem of poincaré's periodic operator in the dfferent phase spaces. A series of simple criteria for the existence, uniqueness and stability of these systems are obtained.

In this paper, a class of reaction diffusion processes with general reaction rates is studied. A necessary and sufficient condition for the reversibility of this calss of reaction diffusion processes is given, and then the ergodicity of these processes is proved.

For any natural numbers m and n≥17 we can construct explicitly indecomposable definite unimodular normal Hermitian lattices of rank n over the ring of algebraic integers R_m in an imaginary quadratic field Q(√-m. It is proved that for any n (in case m=11, there is one exception n=3) there exist indecomposable definite unimodular normal Hermitian R₁₅(R₁₁)-lattices of rank n, and we exhibit representatives for each class. In the exceptional case there are no lattices with the desired properties. The method given in this paper can solve completely the problem of constructing indecomposable definite unimodular normal Hermitian R_m -lattices of any rank n for each m.

In this paper, we deal with the problem of uniqueness of entire or meromorphic functions and obtain some results that are improvements over those of M. Ozawa, H. Ueda, K. Shibazaki and Yi Hongxun. An example shows that the results in this paper are sharp.

A complete characterization of normal operators which are similar to irreducible operators and some related results are given.

This paper provides a polyhedral theory on graphs from which the criteria of Whitney and MacLane for the planarity of graphs are unified, and a brief proof of the Gauss crossing conjecture is obtained.

We give a treatment of the Weiertrass points of curves which is a little different from the treatment by Laksov. We introduce the notion of the ith weight which makes the treatment easier and gives an algorithm for computing the gap sequence of an effective divisor and the weight at a point.

The Steinberg group over a ring along an arbitrary set is defined. Its properties, structure, and isomorphism theory are studied.

Let C be an n-dimensional sphere with diameter 1 and center at the origin in Eⁿ . The view-obstruction problem for n-dimensional spheres is to determine a constant ν(n) to be the lower bound of those α for which any half-line L, given by x_{em = aem} t (em=1,2,…, n) where parameter t≥0 and a_em (em=1,2,……, n) are positive real numbers, intersects
 .
In this paper, for n=3, the following result is proved. For α > we have that any half-line L, given by x_em = a_em t(em=1,2,3), intersects Δ(C,α), where parameter t≥0 and a_em (em=1,2,3) are positive real numbers such that |a|+|b|+|c|≠3 whenever aa₁+ba₂+ca₃=0 for three integers a,b,c.

Let f(z) be an entire function of order λ and of finite lower order μ. If the zeros of f(z) accumulate in the vicinity of a finite number of rays, then
(a) λ is finite;
(b) for every arbitrary number k₁>1, there exists k₂>1 such that T(k₁ r,f)≤k₂ T(r,f) for all r≥ r₀.

The relation between continued fractions and Berlekamp's algorithm was studied by some researchers. The latter is an iterative procedure proposed for decoding BCH codes. However, there remains an unanswered question wheter each of the iterative steps in the algorithm can be interpreted in terms of continued fractions. In this paper, we first introduce the so-called refined convergents to the continued fraction expansion of a binary sequence S, and then give a thorough answer to the question in the context of Massey's linear feedback shift register synthesis algorithm which is equivalent to that of Berlekamp, and at last we prove that there exists a one-to-one correspondence between the n-th refined convergents and the length n segments.

Let D={z∈Σ:|z|<1} and ψ be a normal function on [0,1). For p∈(0,1) such a function ψ is used to define a Bergman space A^p (ψ) on D with weight ψ^p (|·|)/(1-|·|²). In this paper, the dual space of A^p (ψ) is given, four characteristics of Carleson measure on A^p (ψ) are obtained. Moreover, as an application, three sequence interpolation theorems in A^p (ψ) are derived.

In this paper, it is proved that the two real analytic expanding endomorphisms of the unit circle are equivalent if and only if they have equal eigenvalues along corresponding cycles. A sufficient and necessary condition that a real analytic solution of Cvitanovic-Feigenbaum equation induces a real analytic expanding map is given.

In this paper, we completely determine the spectrum of a compact composition operator on lummer-Hardy space.

This paper continues the researches of Yang Lo and others, and gives a further generalization of Gu's criterion for the normality of a family of meromorphic functions.

In this paper, we gave a proof for the local continuity modulus theorem of the Wiener process, i.e.,
a.s.
This result was given by Csérgé and Révész (1981), but the proof gets them nowhere. We also gave a similar local continuity modulus result for the infinite dimensional OU processes.

Let R be a ring with an identity element. R∈IBN means that R^m≌Rⁿ implies m=n, R∈IBN₁ means that R^m≌Rⁿ⊕K implies m≥n, and R∈IBN₂ means that R^m≌R^m⊕K implies K=0. In this paper we give some characteristic properties of IBN₁ and IBN₂, with orderings on the Grothendieck groups. In addition, we obtain the following results: (1) If R∈IBN₁ and all finitely generated projective left R-modules are stably free, then the Grothendieck group K₀(R) is a totally ordered abelian group. (2) If the pre-ordering of the Grothendieck group K₀(R) of a ring R is a partial ordering, then R∈IBN₁ or K₀(R)=0.

In this paper, we study the existence of quasi-periodic solutions and the boundedness of solutions for a wide class nonlinear differential equations of second order. Using the KAM theorem of reversible systems and the theory of transformations we obtain the existence of quasi-periodic solutions and the boundedness of solutions under some reasonable conditions.

In this paper we investigate whether a polynomial algebra can be realized as a cohomology ring of a topological space. Our main results are that we can split the realizable polynomial algebra into a tensor product of certain simple factors and that these factors are given explicitly when p>7. What is worth mentioning is that most of these factors are known to be realizable.

In this paper, the Bers-Orlicz spaces on the automorphic form A_α^φ (G) (or EA_α^φ (G)) and L_α^φ(G) on the product Riemann surfaces are studied. We prove that each f∈A_α^φ (G) is a cusp form. For f∈A_α^φ (G), we give the reproducing formula. And, we give the projective operator P_a from L_α^φ (G) to A_α^φ (G) to EA_α^φ (G)). After giving some fundamental properties of the Poincaré series, we prove a dual theorem A_α^φ* (G)=(EA_α^φ(G))^*.

This paper is devoted to studying a new topic: optimal Markovian couplings, mainly for time-continuous Markov processes. The study emphasizes the analysis of the coupling operators rather than the processes. Some constructions of optimal Markovian couplings for Markov chains and diffusions are presented, which are often unexpected. Then, the results are applied to study the L²-convergence for Markov chains and for a diffusion on compact manifold. The estimate of the convergent rate provided by this method can be sharp.

Consider the nonparametric regression model Y= g₀(T)+ u, where Y is real-valued, u is a random error, T ranges over a nondegenerate compact interval, say [0,1], and g₀(·) is an unknown regression function, which is m(m≥0) times continuously differentiable and its mth derivative, g₀^(m), satisfies a Hölder condition of order γ(m+γ>1/2). A piecewise polynomial L₁-norm estimator of g₀ is proposed. Under some regularity conditions including that the random errors are independent but not necessarily have a common distribution, it is proved that the rates of convergence of the piecewise polynomial L₁-norm estimator are almost surely and in probability, which can arbitrarily approach the optimal rates of convergence for nonparametric regression, where σ is any number in (0, min((m+γ-1/2)/3,γ)).

The Hausdorff dimensions of the image and the graph of random fields are given under general conditions. The results can be used to a wider class of self-similar random fields and processes, including Brownian motion, Brownian sheet, fractional Brownian motion, processes with stable or (α, β)-fractional stable components.

This paper introduced a new homotopy extension property called the fairly weak homotopy extension property (FWHEP) and showed some its characterizations and properties. The FWHEP is strictly between the WHEP and the VWHEP (RWHEP).

This paper is to study the characteristic functions of meromorphic functions for an angular domain. We show an inequality which is similar to the inequality in the second fundamental theorem of Nevanlinna, and estimate its remainder. Finally, as an application we obtain a sufficient and necessary condition of Borel directions.

This paper discusses some problems on the cardinal spline interpolation corresponding to infinite order differential operators. The remainder formulas and a dual theorem are established for some convolution classes, where the kernels are PF densities. Moreover, the exact error of approximation of a convolution class with interpolation cardinal splines is determined. The exact values of average n-Kolmogorov widths are obtained for the convolution class.

We discuss the existence of global classical solution for the uniformly parabolic equation

Where a is strongly nonlinear with respect to u_xx and φ is not necessarily small. We also deal with nonuniform case.

Let X₁,X₂,...,X_n be a sequence of nonnegative independent random variables with a common continuous distribution functionF.Let Y₁,Y₂,...,Y_n be another sequence of nonnegative independent random variables with a common continuous distribution function G,also independent of {X_i}.We can only observe Z_i=min(X_i,Y_i),and .Let H=1-(1-F)(1-G) be the distribution function of Z.In this paper,the limit theorems for the ratio of the Kaplan-Meier estimator Ŝ_n(t) or the Altshuler estimator S_n(t) to the true survival function S(t) are given.It is shown that (1)P(δ_(n)=1 i.o.)=0 if F(τ_H)<1 and P(δ_n=0 i.o.)=0 ifG(τH) > 1 where δ_(n) is the corresponding indicator function of and have the same order a.s.,where {T_n} is a sequence of constants such that 1-H(T_n)=n^-α(logn)^β(log logn)^γ.