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1.
The authors establish sufficient conditions for the oscillation of all solution of neutral delay differential equations of even and odd order of the form $$\[\frac{{{d^n}}}{{d{t^n}}}[y(t) + p(t)y(t - \tau )] + Q(t)y(t - C) = 0,t \ge {t_0},\]$$ where $\[P,Q \in C[[{t_0},\infty ),R],\tau ,\sigma \in {R^ + }\]$ and $\[n \ge 1\]$.  相似文献   

2.
Based on [3] and [4],the authors study strong convergence rate of the k_n-NNdensity estimate f_n(x)of the population density f(x),proposed in [1].f(x)>0 and fsatisfies λ-condition at x(0<λ≤2),then for properly chosen k_nlim sup(n/(logn)~(λ/(1 2λ))丨_n(x)-f(x)丨C a.s.If f satisfies λ-condition,then for propeoly chosen k_nlim sup(n/(logn)~(λ/(1 3λ)丨_n(x)-f(x)丨C a.s.,where C is a constant.An order to which the convergence rate of 丨_n(x)-f(x)丨andsup 丨_n(x)-f(x)丨 cannot reach is also proposed.  相似文献   

3.
This paper studios the existence of closed geodesics in the homotopy class of a given closed curve. Let M be a complete Riemannian manifold without boundary, f∈C~1(S~1, M). Look at S~1 as [0, 2π]/{0, 2π}. The following results are proved:A. The initial value problem of heat equation _if_t=τ(f_i), f_0=f always admits a global solution.B. (Existence of closed geodesics). If there exists a compact set KM such that f(S~1)∩K≠φ andE(f)≤(1/π)l(K)~2,then there exists a closed geodesic homotopic to f. If then the closed geodesic is minimal.C. Some estimates about injective radius are obtained.Some example is found showing that the inequalities in B are sharp.  相似文献   

4.
MINIMAL SUBMANIFOLDS IN A RIEMANNIAN MANIFOLD OF QUASI CONSTANT CURVATURE   总被引:8,自引:0,他引:8  
A Riemannian manifold V~m which admits isometric imbedding into two spaces V~(m+p)ofdifferent constant curvatures is called a manifold of quasi constant curvature.TheRiemannian curvature of V~m is expressible in the formand conversely.In this paper it is proved that if M~n is any compact minimal submanifoldwithout boundary in a Riemannian manifold V~(n+p)of quasi constant curvature,then∫_(M~u)(2-1/p)σ~2-[na+1/2(b-丨b丨)(n+1)]σ+n(n-1)b~2*丨≥0,where σ is the square of the norm of the second fundamental form of M~n When V~(n+p)is amanifold of constant curvature,b=0,the above inequality reduces to that of Simons.  相似文献   

5.
Let X_1,…,X_n be a sequence of independent identically distributed random variableswith distribution function F and density function f.The X_are censored on the right byY_i,where the Y_i are i.i.d.r.v.s with distribution function G and also independent of theX_i.One only observesLet S=1-F be survival function and S be the Kaplan-Meier estimator,i.e.,where Z_are the order statistics of Z_i and δ_((i))are the corresponping censoring indicatorfunctions.Define the density estimator of X_i by where =1-and h_n(>0)↓0.  相似文献   

6.
AIn this paper, the author obtains the following results:(1) If Taylor coeffiients of a function satisfy the conditions:(i),(ii),(iii)A_k=O(1/k) the for any h>0 the function φ(z)=exp{w(z)} satisfies the asymptotic equality the case h>1/2 was proved by Milin.(2) If f(z)=z α_2z~2 …∈S~* and,then for λ>1/2  相似文献   

7.
In this paper,, the author proves the following result: Let $\[{E_{a,k}}(N)\]$ denote the number of natural numbers $\[n \le N\]$ for which equation $$\[\sum\limits_{i = 0}^k {\frac{1}{{{x_i}}}} = \frac{a}{n}\]$$ is insolable in positive integers $\[{x_i}(i = 0,1, \cdots ,k)\]$.Then $$\[{E_{a,k}}(N) \ll N\exp \{ - C{(\log N)^{1 - \frac{1}{{k + 1}}}}\} \]$$ where the implied constant depends on a and K.  相似文献   

8.
Let M~n (n≥2) be a complex Kaehler submanifold immersed in the complex projective space CP~m(1). Let K be the sectional curvature of M~n. Then K≥1/8 if and only if M~n is an imbedding submanifold congruent to the standard imbedding CP~n (1) or CP~n(1/2).  相似文献   

9.
The number $\[A({d_1}, \cdots ,{d_n})\]$ of solutions of the equation $$\[\sum\limits_{i = 0}^n {\frac{{{x_i}}}{{{d_i}}}} \equiv 0(\bmod 1),0 < {x_i} < {d_i}(i = 1,2, \cdots ,n)\]$$ where all the $\[{d_i}s\]$ are positive integers, is of significance in the estimation of the number $\[N({d_1}, \cdots {d_n})\]$ of solutiohs in a finite field $\[{F_q}\]$ of the equation $$\[\sum\limits_{i = 1}^n {{a_i}x_i^{{d_i}}} = 0,{x_i} \in {F_q}(i = 1,2, \cdots ,n)\]$$ where all the $\[a_i^''s\]$ belong to $\[F_q^*\]$. the multiplication group of $\[F_q^{[1,2]}\]$. In this paper, applying the inclusion-exclusion principle, a greneral formula to compute $\[A({d_1}, \cdots ,{d_n})\]$ is obtained. For some special cases more convenient formulas for $\[A({d_1}, \cdots ,{d_n})\]$ are also given, for example, if $\[{d_i}|{d_{i + 1}},i = 1, \cdots ,n - 1\]$, then $$\[A({d_1}, \cdots ,{d_n}) = ({d_{n - 1}} - 1) \cdots ({d_1} - 1) - ({d_{n - 2}} - 1) \cdots ({d_1} - 1) + \cdots + {( - 1)^n}({d_2} - 1)({d_1} - 1) + {( - 1)^n}({d_1} - 1).\]$$  相似文献   

10.
Suppose that there is a variance components model $$\[\left\{ {\begin{array}{*{20}{c}} {E\mathop Y\limits_{n \times 1} = \mathop X\limits_{n \times p} \mathop \beta \limits_{p \times 1} }\{DY = \sigma _2^2{V_1} + \sigma _2^2{V_2}} \end{array}} \right.\]$$ where $\[\beta \]$,$\[\sigma _1^2\]$ and $\[\sigma _2^2\]$ are all unknown, $\[X,V > 0\]$ and $\[{V_2} > 0\]$ are all known, $\[r(X) < n\]$. The author estimates simultaneously $\[(\sigma _1^2,\sigma _2^2)\]$. Estimators are restricted to the class $\[D = \{ d({A_1}{A_2}) = ({Y^''}{A_1}Y,{Y^''}{A_2}Y),{A_1} \ge 0,{A_2} \ge 0\} \]$. Suppose that the loss function is $\[L(d({A_1},{A_2}),(\sigma _1^2,\sigma _2^2)) = \frac{1}{{\sigma _1^4}}({Y^''}{A_1}Y - \sigma _1^2) + \frac{1}{{\sigma _2^4}}{({Y^''}{A_2}Y - \sigma _2^2)^2}\]$. This paper gives a necessary and sufficient condition for $\[d({A_1},{A_2})\]$ to be an equivariant D-asmissible estimator under the restriction $\[{V_1} = {V_2}\]$, and a sufficient condition and a necessary condition for $\[d({A_1},{A_2})\]$ to equivariant D-asmissible without the restriction.  相似文献   

11.
This paper deals with the following IBV problem of nonlinear parabolic equation: $$\[\left\{ {\begin{array}{*{20}{c}} {{u_t} = \Delta u + F(u,{D_x}u,D_x^2u),(t,x) \in {B^ + } \times \Omega ,}\{u(0,x) = \varphi (x),x \in \Omega }\{u{|_{\partial \Omega }} = 0} \end{array}} \right.\]$$ where $\[\Omega \]$ is the exterior domain of a compact set in $\[{R^n}\]$ with smooth boundary and F satisfies $\[\left| {F(\lambda )} \right| = o({\left| \lambda \right|^2})\]$, near $\[\lambda = 0\]$. It is proved that when $\[n \ge 3\]$, under the suitable smoothness and compatibility conditions, the above problem has a unique global smooth solution for small initial data. Moreover, It is also proved that the solution has the decay property $\[{\left\| {u(t)} \right\|_{{L^\infty }(\Omega )}} = o({t^{ - \frac{n}{2}}})\]$, as $\[t \to + \infty \]$.  相似文献   

12.
In this paper, we consider the relative position of limit cycles for the system $$\[\begin{array}{*{20}{c}} {\frac{{dx}}{{dt}} = \delta x - y + mxy - {y^2}}\{\frac{{dy}}{{dt}} = x + a{x^2}} \end{array}\]$$ under the condition $$\[a < 0,0 < \delta \le m,m \le \frac{1}{a} - a\]$$ The main result is as follows: (i)Under Condition (2), if $\[\delta = \frac{m}{2} + \frac{{{m^2}}}{{4a}} \equiv {\delta _0}\]$, then system $\[{(1)_{{\delta _0}}}\] $ has no limit cycles and on singular closed trajectory through a saddle point in the whole plane, (ii)Under condition (2), the foci 0 and R'' cannot be surrounded by the limit cycles of system (1) simultaneously.  相似文献   

13.
Let L(x) denote the number of square-full integers not exceeding x. It is proved in [1] thatL(x)~(ζ(3/2)/ζ(3))x~(1/2) (ζ(2/3)/ζ(2))x~(1/3) as x→∞,where ζ(s) denotes the Riemann zeta function. Let △(x) denote the error function in the asymptotic formula for L(x). It was shown by D. Suryanaryana~([2]) on the Riemann hypothesis (RH) that1/x integral from n=1 to x |△(t)|dt=O(x~(1/10 s))for every ε>0. In this paper the author proves the following asymptotic formula for the mean-value of △(x) under the assumption of R. H.integral from n=1 to T (△~2(t/t~(6/5))) dt~c log T,where c>0 is a constant.  相似文献   

14.
Consider the nonlinear inltial-boundary value problem for quasilinear hyperbolicsystem:Let k≥2[n/2] 6,(F,g)∈ H~k(R_ ;Ω)×H~k(R_ ;Ω),and their traces at t=0 are zeroup to the order k-1.If for u=0,the problem(*)at t=0 is a Kreiss hyperbolic system,and the boundaryconditions satisfy the uniformly Lopatinsky criteria,then there exists a T>0 such that(*)has a unique H~k soluton in(0,T).In the Appendix,for symmetric hyperbolic systems,a comparison between theuniformly Lopatinsky condition and the stable admissible condition is given.  相似文献   

15.
16.
Consider the higher-order neutral delay differential equationd~t/dt~n(x(t)+sum from i=1 to lp_ix(t-τ_i)-sum from j=1 to mr_jx(t-ρ_j))+sum from k=1 to Nq_kx(t-u_k)=0,(A)where the coefficients and the delays are nonnegative constants with n≥2 even. Then anecessary and sufficient condition for the oscillation of (A) is that the characteristicequationλ~n+λ~nsum from i=1 to lp_ie~(-λτ_i-λ~n)sum from j=1 to mr_je~(-λρ_j)+sum from k=1 to Nq_ke~(-λρ_k)=0has no real roots.  相似文献   

17.
By using the exponential dichotomy and the averaging method,a perturbation theoryis established for the almost periodic solutions of an almost differential system.Suppose that the almost periodic differential system(dx)/(dt)=f(x,t) ε~2g(x,t,ε)(1)has an almost periodic solution x=x_0(t,M)for ε=0,where M=(m_1,…,m_k)is theparameter vector.The author discusses the conditions under which(1)has an almostperiodic solution x=x(t,ε)such that x(t,ε)=x_0(t,M)holds uniformly.The results obtained are quite complete.  相似文献   

18.
In the present paper, we show that there exist a bounded, holomorphic function $\[f(z) \ne 0\]$ in the domain $\[\{ z = x + iy:\left| y \right| < \alpha \} \]$ such that $\[f(z)\]$ has a Dirichlet expansion $\[\sum\limits_{n = 0}^{ + \infty } {{d_n}{e^{ - {u_n}}}} \]$ in the halfplane $\[x > {x_f}\]$ if and only if $\[\frac{a}{\pi }\log r - \sum\limits_{{u_n} < r} {\frac{2}{{{u_n}}}} \]$ has a finite upperbound on $\[[1, + \infty )\]$, where $\[\alpha \]$ is a positive constant,$\[{x_f}( < + \infty )\]$ is the abscissa of convergence of $\[\sum\limits_{n = 0}^{ + \infty } {{d_n}{e^{ - {u_n}}}} \]$ and the infinite sequence $\[\{ {u_n}\} \]$ satisfies $\[\mathop {\lim }\limits_{n \to + \infty } ({u_{n + 1}} - {u_n}) > 0\]$. We also point out some necessary conditions and sufficient ones Such that a bounded holomorphic function in an angular(or half-band) domain is identically zero if an infinite sequence of its derivatives and itself vanish at some point of the domain. Here some result are generalizations of those in [4].  相似文献   

19.
The paper proves on the basis of [1] the following theorem: Let $\[f(z)\]$ be an entire function of lower order $\[\mu < \infty \]$, and $\[{a_i}(z)(l = 1,2, \cdots ,k)\]$ be meromorphic functions which satisfy $\[T(r,{a_i}(z)) = o\{ T(r,f)\} \]$. If $$\[\sum\limits_{i = 1}^k {\delta ({a_i}(z),f) = 1\begin{array}{*{20}{c}} {({a_i}(z) \ne \infty )}&{(1)} \end{array}} \]$$ then the deficiencies $\[\delta ({a_i}(z),f)\]$ are equal to $\[\frac{{{n_1}}}{\mu }\]$, where $\[{n_i}\]$ is an integer,$\[l = 1,2, \cdots ,k\]$.  相似文献   

20.
Let $V$ be a non-defective S-dimensional quadratic space over a field $F$ of characteristic 2, $\[F \ne {F_2}\]$. We prove that if there is an exceptional automorphism of either $\[{\Omega _S}(V)\]$ or $\[O_S^''(V)\]$ then $\[{V^\alpha }\]$ has a Cayley algebra structure for some $\[\alpha \]$ in F. Moreover, every exceptional automorphism of $\[O_S^''(V)\]$ has exactly one of the following forms: $$\[{\varphi _1} \circ {\Phi _g}or{\varphi _2} \circ {\Phi _g}\]$$ where $\[{\Phi _g}\]$ is an automorphism of $\[O_S^''(V)\]$ given by conjugation by a semilinear automorphism of V which preserves the quadratic structure, and $\[{\varphi _1}\]$ and $\[{\varphi _2}\]$ are the automorphisms induced by triality principle. Every exceptional automorphism of $\[{\Omega _S}(V)\]$ is the restriction of a unique exceptional automorpliism of $\[O_S^''(V)\]$.  相似文献   

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