脉冲泛函微分方程的渐近性态

该文讨论脉冲泛函微分方程$\left\{\begin{array}{ll}x^,(t)=f(t,x_t), t≥ t₀,△x=I_k(t,x(t^-)), t=t_k,k∈ Z⁺,给出了方程零解渐近稳定性和一致渐近稳定性的充分条件,指出这些条件推广或改进了文献[7--9]的相应结论.     

具有非正系数的一阶中立型滞后微分方程的振动性和渐近性质

本文对较文[1,2]中更为广泛的具有非正系数的一类线性方程d/(dt)[x(t)+p(t)x(t-τ)]-Q(t)x(t-σ(t))=0,t≥t₀及非线性方程d/(dt)[x(t)+p(t)x(t-τ(t))]-Q(t)f(x(t-σ(t)))=0,t≥t₀进行了讨论,其中Q(t)∈C([t₀,+∞).R₊),得到了保证上述方程的所有有界解振动及非振动解当t→+∞时趋于零或±∞的一些充分性准则.     

具投放的中立型时滞竞争扩散系统正周期解的存在性

利用Nussbaum 度理论建立了具投放的中立型时滞竞争扩散系统 x′₁(t)=x₁(t)[a₁(t)-b₁(t)x₁(t)-c₁(t)y(t) ]+D₁(t)[x₂(t)-x₁(t)]+S₁(t), x′₂(t)=x₂(t)[a₂(t)-b₂(t)x₂(t)]+D₂(t)[x₁(t)-x₂(t) ]+S₂(t), y′(t)=y(t)[a₃(t)-b₃(t)y(t)-α(t)y(t-τ₁(t))-β(t) ∫⁰_τ k(s)y(t+s)ds -γ(t)y’(t-τ₂(t))-c₃(t)x₁(t)]. 存在正周期解的一个充分条件.     

广义 Lévy单的样本轨道性质

设X(t)是下指数为α取值于R^d的N参数广义Lévy 单, R={(x,t]=∏^N_i=1 (s_i,t_i], s_i<t_i}, E(x, Q)={t∈Q: X(t)=x}, Q∈∏, 是 X在点x处的水平集, X(Q)={x: 设X(t)是下指数为α取值于R^d的N参数广义Lévy 单, R={(x,t]=∏^N_i=1 (s_i,t_i], s_i<t_i}, E(x, Q)={t∈Q: X(t)=x}, Q∈∏, 是 X在点x处的水平集, X(Q)={x: 设X(t)是下指数为α取值于Rd的N参数广义Lévy单,R={(s,t]=∏Ni=1(si,ti],si＜ti},E(x,Q)={t∈Q∶X(t)=x},Q∈R,是X在点x处的水平集,X(Q)={x∶(∈)t∈Q,使得X(t)=x}为X在Q上的像集.本文探讨了X(t)局部时存在性及其增量的大小.同时,也得到了水平集E(x,Q)Hausdorff维数和X(Q)一致维数上界的结果.     

POSITIVE SOLUTION TO FOURTH-ORDER IMPULSIVE DIFFERENTIAL EQUATIONS WITH P-LAPLACIAN

In order to study three-point BVPs for fourth-order impulsive differential equation of the form(\phi_p(u'(t)))'- f(t,u(t))=0, t≠ t_i,△ u(t_i)=-I_i(u(t_i)), i=1, 2, ..., k,△u'(t_i)=-L_i(u(t_i)), i=1, 2, ..., k,(\star)with the following boundary conditionsu'(0)=u(1)=0, u'(0)=0=u'(1)-\phi_q(α)u'(η),the authors translate the fourth-order impulsive differential equations with p-Laplacian (\star) into three-point BVPs for second-order differential equation without impulses and two-point BVPs for second-order impulsive differential equation by a variable transform. Based on it, existence theorems of positive solutions for the boundary value problems (\star) are obtained.     

有理函数域上的Gross猜想

研究分圆函数域扩张k(Λ_f)/k情形下的Gross猜想, 其中k=F_q(t)是有理函数域, f是k上的首一多项式.通过直接计算,证明了在Fermat曲线(即f=t(t&#8722;1))情形时猜想成立.当f为不可约多项式时,证明了Gross猜想和Weil互反律等价.对一般情形,证明了弱Gross猜想成立.     

构造k紧优双环网的无限族的新方法

双环网(double loop network)是具有n个结点和出度为2的有向循环图, 已广泛地应用于局域网和分布系统的设计中. 给出了构造k紧优双环网的无限族的新方法,对于k=0,1,…,40,用此方法可构造k紧优双环网的无限族, 其中结点数n_k(t,a) 是t的二次多项式且含有参数a; 并提出了一个猜想.     

具有“积分小”系数的中立型方程的振动性

讨论了中立型方程d/(dt)[x(t) - R(t)x(t - r)] + P(t)x(t - τ) - Q(t)x(t - δ) = 0,的振动性,其中P,Q,R∈C（[t₀,∞), R⁺）,r,τ,δ∈（0,∞）,得到若干新结果。     

一类二阶泛函微分方程的多重周期解

利用变分原理和Z_2不变群指标,得出二阶泛函微分方程x″(t-τ)+c(x(t)-x(t-2τ))′-x(t-τ)+λf(t,x(t),x(t-τ),x(t-2τ))=0的多重周期解的存在性质.     

定义在迭代函数系统吸引子上的动力系统的平衡态

在该文中, 令E表示一个迭代函数系统(X,T₁,…, T_m). 的吸引子. 定义连续自映射 f : E→E为f(x)=T^-1_j(x), x∈ T_j(E), j=1, …, m . 给定Given ψ ∈C_R(E), 令 K_ψ(δ, n = sup{∣∑^n-1_k=0[ψ(f ^kx)-ψ(f ^ky)]|:y ∈ B_x (δ, n)}, 这里B_x(δ, n) 表示Bowen球. 取一个扩张常数 ε, 记K_ψ=sup_n K_ψ(ε, n) , 定义ν(E)={ψ : K_ψ < ∞}. 对f : E → E, 作为Ruelle的一个定理^{[3, 定理2.1]}的一个应用, 我们证明每个ψ ∈ν(E)具有惟一的平衡态. 此结果推广了文献[12]中的主要结果.     

The Betti Numbers of the Free Loop Space of a Connected Sum

Let M₁ and M₂ be two simply connected closed manifolds of thesame dimension. It is proved that (1) if k is a coefficient field such that neither M₁ nor M₂has the same cohomology as a sphere, then the sequence (b_k)_k1of Betti numbers of the free loop space on M₁ #M₂ is unbounded; (2) if, moreover, the cohomology H*(M₁;k) is not generated asalgebra by only one element, then the sequence (b_k)_k1 has anexponential growth. Thanks to theorems of Gromoll and Meyer and of Gromov, thisimplies, in case 1, that there exist infinitely many closedgeodesics on M₁#M₂ for each Riemannian metric, and, in case2, that for a generic metric, the number of closed geodesicsof length t grows exponentially with t.     

On Simultaneous Diagonal Inequalities

Let F₁, ..., F_t be diagonal forms of degree k with real coefficientsin s variables, and let be a positive real number. The solubilityof the system of inequalities |F₁(x)|<,...,|F_t(x)|< in integers x₁, ..., x_s has been considered by a number of authorsover the last quarter-century, starting with the work of Cook[9] and Pitman [13] on the case t = 2. More recently, Brüdernand Cook [8] have shown that the above system is soluble providedthat s is sufficiently large in terms of k and t and that theforms F₁, ..., F_t satisfy certain additional conditions. Whathas not yet been considered is the possibility of allowing theforms F₁, ..., F_t to have different degrees. However, with therecent work of Wooley [18,20] on the corresponding problem forequations, the study of such systems has become a feasible prospect.In this paper we take a first step in that direction by studyingthe analogue of the system considered in [18] and [20]. Let₁, ..., _s and µ₁, ..., µ_s be real numbers such thatfor each i either _i or µ_i is nonzero. We define the forms and consider the solubility of the system of inequalities in rational integers x₁, ..., x_s. Although the methods developedby Wooley [19] hold some promise for studying more general systems,we do not pursue this in the present paper. We devote most ofour effort to proving the following theorem.     

Complex Monodromy and Changing Real Pictures

The purpose of this note is to generalise, and give a more illuminatingproof, of a theorem of [13] (Theorem 1.1 below). Before statingit, we provide some introductory information. Consider the followingtwo sequences of pictures: in each we see a 1-parameter familyX_R,t of real algebraic hypersurfaces, which undergoes a bifurcationwhen the parameter t is equal to 0. Note that in Figure 1, both(i) (a) and (i) (b), and in (ii) (b), the surface X_R,t has apurely 1-dimensional part, which we have indicated with a dottedline, and that in (i) (b) we have drawn a curve vertically alongthe middle of the surface to make clearer the way it passesthrough itself. The reader will observe that in (a) the surfaceX_R,t is homotopically a 2-sphere when t>0 and a 0-spherewhen t<0, while in (b) X_R,t is a homotopy 1-sphere both fort<0 and t>0. Such sequences are typical in singularity theory; each is infact the family of algebraic closures of images of a versaldeformation of a codimension 1 singularity of mapping. Now suppose that the complexification X_C,t is a homotopy n-sphere.In [13] the second author pointed out that it follows that X_R,tis a homotopy sphere for t0 (allowing the empty set as a –1-sphere).Indeed, in the local situation, or globally in the weightedhomogeneous case, there are well-defined integers k₊ and k_–between –1 and n such that X_R,tS^k+ for t>0 and X_R,tS^k–for t<0. We describe X_R,t for tR–0 as ‘good’ if thehomotopy dimension of X_R,t is equal to n. In this case the inclusionX_R,tX_t is a homotopy equivalence [13, 1.1].     

A Necessary and Sufficient Condition for a Linear Differential System to be Strongly Monotone

In order to present the results of this note, we begin withsome definitions. Consider a differential system [formula] where IR is an open interval, and f(t, x), (t, x)IxRⁿ, is acontinuous vector function with continuous first derivativesf_r/x_s, r, s=1, 2, ..., n. Let D_xf(t, x), (t, x)IxRⁿ, denote the Jacobi matrix of f(t,x), with respect to the variables x₁, ..., x_n. Let x(t, t₀,x₀), tI(t₀, x₀) denote the maximal solution of the system (1)through the point (t₀, x₀)IxRⁿ. For two vectors x, yRⁿ, we use the notations x>y and x>>yaccording to the following definitions: [formula] An nxn matrix A=(a_rs) is called reducible if n2 and there existsa partition [formula] (p1, q1, p+q=n) such that [formula] The matrix A is called irreducible if n=1, or if n2 and A isnot reducible. The system (1) is called strongly monotone if for any t₀I, x₁,x₂Rⁿ [formula] holds for all t>t₀ as long as both solutions x(t, t₀, x_i),i=1, 2, are defined. The system is called cooperative if forall (t, x)IxRⁿ the off-diagonal elements of the nxn matrix D_xf(t,x) are nonnegative. 1991 Mathematics Subject Classification34A30, 34C99.     

Sandwiching C0-Semigroups

Let T = {T(t)}_t0 be a C₀-semigroup on a Banach space X. Thefollowing results are proved. (i) If X is separable, there exist separable Hilbert spacesX₀ and X₁, continuous dense embeddings j₀:X₀ X and j₁:X X₁,and C₀-semigroups T₀ and T₁ on X₀ and X₁ respectively, suchthat j₀ T₀(t) = T(t) j₀ and T₁(t) j₁ = j₁ T(t) for all t 0. (ii) If T is -reflexive, there exist reflexive Banach spacesX₀ and X₁ , continuous dense embeddings j:D(A²) X₀, j₀:X₀ X, j₁:X X₁, and C₀-semigroups T₀ and T₁ on X₀ and X₁ respectively,such that T₀(t) j = j T(t), j₀ T₀(t) = T(t) j₀ and T(t) j₁ = j₁ T(t) for all t 0, and such that (A₀) = (A) = (A₁),where A_k is the generator of T_k, k = 0, Ø, 1.     

具连续变量脉冲差分方程解的振动性

考虑新的一类具有连续变量的脉冲差分方程x(t τ) - x(t) p(t)x(t - rτ) =0,x(tk τ) - x(tk) = bkx(tk),　t≥t0 -τ,t≠tk,t∈N(1),其中p(t)是[t0 -τ,∞]上的非负连续函数,τ>0,bk 是常数,r是正整数, 0≤t0 < t1 < t2 <…< tk <…且limk→∞tk =∞,获得了方程所有解振动的充分条件.     

On Waring's Problem in Number Fields

Let K be an algebraic number field of degree n over the rationals,and denote by J_k the subring of K generated by the kth powersof the integers of K. Then G_K(k) is defined to be the smallests1 such that, for all totally positive integers vJ_k of sufficientlylarge norm, the Diophantine equation (1.1) is soluble in totally non-negative integers _i of K satisfying N(_i)<<N(v)^1/k (1is). (1.2) In (1.2) and throughout this paper, all implicit constants areassumed to depend only on K, k, and s. The notation G_K(k) generalizesthe familiar symbol G(k) used in Waring's problem, since wehave G_Q(k) = G(k). By extending the Hardy–Littlewood circle method to numberfields, Siegel [8, 9] initiated a line of research (see [1–4,11]) which generalized existing methods for treating G(k). Thistypically led to upper bounds for G_K(k) of approximate strengthnB(k), where B(k) was the best contemporary upper bound forG(k). For example, Eda [2] gave an extension of Vinogradov'sproof (see [13] or [15]) that G(k)(2+o(1))k log k. The presentpaper will eliminate the need for lengthy generalizations assuch, by introducing a new and considerably shorter approachto the problem. Our main result is the following theorem.     

On Artin's Conjecture, I: Systems of Diagonal Forms

As a special case of a well-known conjecture of Artin, it isexpected that a system of R additive forms of degree k, say [formula] with integer coefficients a_ij, has a non-trivial solution inQ_p for all primes p whenever [formula] Here we adopt the convention that a solution of (1) is non-trivialif not all the x_i are 0. To date, this has been verified onlywhen R=1, by Davenport and Lewis [4], and for odd k when R=2,by Davenport and Lewis [7]. For larger values of R, and in particularwhen k is even, more severe conditions on N are required toassure the existence of p-adic solutions of (1) for all primesp. In another important contribution, Davenport and Lewis [6]showed that the conditions [formula] are sufficient. There have been a number of refinements of theseresults. Schmidt [13] obtained N>>R²k³ log k, and Low,Pitman and Wolff [10] improved the work of Davenport and Lewisby showing the weaker constraints [formula] to be sufficient for p-adic solubility of (1). A noticeable feature of these results is that for even k, onealways encounters a factor k³ log k, in spite of the expectedk² in (2). In this paper we show that one can reach the expectedorder of magnitude k². 1991 Mathematics Subject Classification11D72, 11D79.     

A wavelike functional equation of Pexider type

Summary. Let (G, +) and (H, +) be abelian groups such that the equation 2u = v 2u = v is solvable in both G and H. It is shown that if f₁, f₂, f₃, f₄, : G ×G ? H f_1, f_2, f_3, f_4, : G \times G \longrightarrow H satisfy the functional equation f₁(x + t, y + s) + f₂(x - t, y - s) = f₃(x + s, y - t) + f₄(x - s, y + t) for all x, y, s, t ? G x, y, s, t \in G , then f₁, f₂, f₃, and f₄ are given by f₁ = w + h, f₂ = w - h, f₃ = w + k, f₄ = w - k where w : G ×G ? H w : G \times G \longrightarrow H is an arbitrary solution of f (x + t, y + s) + f (x - t, y - s) = f (x + s, y - t) + f (x - s, y + t) for all x, y, s, t ? G x, y, s, t \in G , and h, k : G ×G ? H h, k : G \times G \longrightarrow H are arbitrary solutions of D_y,t³g(x,y) = 0 \Delta_{y,t}^{3}g(x,y) = 0 and D_x,t³g(x,y) = 0 \Delta_{x,t}^{3}g(x,y) = 0 for all x, y, s, t ? G x, y, s, t \in G .