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1.
§ 1. Introduction We consider the singular nonlinear boundary value problem where l=v+3/v-1,l+1 is the critical exponent of the embedding of weighted Sobolev space W_t~2~(1,2)(O, +∞) into L_t~2~q(O, ∞), v>2. When v=N-1 we can get the radial solutions of problem where 2~*=2N/N-2 is the critical exponent of the Sobolev embedding H~1(R~n)→L~Q(R~N). Kurtz has discussed the existence of κ-node solution of (1.1), (1.2) for each κ∈N U{0} when the growth rate of |u|~(l-1)u+f(u) is lower then |u|~(v+3/v-1) i.e. 相似文献
2.
THE UNIFORM CONVERGENCE RATE OF KERNEL DENSITY ESTIMATE 总被引:1,自引:1,他引:0
Yang Zhenhai 《数学年刊B辑(英文版)》1985,6(3):335-344
In this paper,we study the uniform convergence rate of kernel density estimate f_nand get optimal uniform rate of convergence without the assumption of compact supportfor kernel function.It is proved that if the density function f satisfies λ-condition andthe kernel function K is λ-good(see section 1),then we havelimsup (n/(logn))~(λ/(1+2λ))丨_n(x)-f(x)丨≤const,a.s. 相似文献
3.
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
4.
Qin Tiehu 《数学年刊B辑(英文版)》1985,6(3):289-298
This paper discusses the following initial-boundary value problems for the first orderquasilinear hyperbolic systems:(u)/(t)+A(u)(u)/(x)=0,(1)u~Ⅱ=F(u~Ⅰ),as x=0,(2)u~Ⅰ=G(u~Ⅱ),as x=L,(3)u=u~0(x),as t=0,(4)where the boundary conditions(2),(3)satisfy F(0)=0,G(0)=0 and the dissipativeconditions,that is,the spectral radii of matrices B_1=(F)/(u~Ⅰ)(0)(G)/(u~Ⅱ)(0)and B_2(G)/(u~Ⅱ)(0)(F)/(u~Ⅰ)(0) are less than unit.Under certain assumptions it is proved that the initial-boundary problem (1)—(4)admits a unique global smooth solution u(x,t)and the C~1-norm丨u(t)丨σ~2of u(x,t)decaysexponentially to zero as t→∞,provided that the C~1-norm丨u~0丨σ~1of the initial data issufficiently small. 相似文献
5.
Yang Zhenai 《数学年刊B辑(英文版)》1990,11(4):536-545
Let be the collection of m-times continuously differentiable probability densities fon R~d such that 丨D~af(x_1)-D~af(x_2)丨≤M‖x_1-x_2‖~β for x_1,x_2∈R~d,[a]=m,where D~adenotes the differential operator defined by D~a=([a])/(x_1~a…x_d~a_d).Under rather weak conditionson K(x),the necessary and sufficient conditions for sup丨_n(x)-f(x)丨=0(((logn/n)~λ/(d+3λ),λ=m+β,f∈ are that ∫x~aK(xi)dx=0 for 0<[a]≤m.Finally the convergenco rate at apoint is given. 相似文献
6.
任德麟 《数学物理学报(B辑英文版)》1984,(1)
Let P_n, be the n-dimensional projective space. Let x_0,x_1,…,x_n be the homogeneous coordinates of point x. Consider the quadric φ(x) =x_0~2+x_1~2+…x_(n-1)~2+εx_n~2, where ε can take the value of +1 or -1. We assume the homogeneous coordinates of the points x that do not belong to φ = 0 are normalized, such that 相似文献
7.
This paper is concerned with the stability of the rarefaction wave for the generalized KdV-Burgers equationRoughly speaking, under the assumption that u_ < u+, the solution u(x,t) to Cauchy problem (1) satisfying sup \u(x,t) -uR(x/t)| -0 as t - , where uR(x/t) is the rarefac-tion wave of the non- viscous Burgers equation ut + f(u)x=0 with Riemann initial data 相似文献
8.
Bai Zhidong 《数学年刊B辑(英文版)》1985,6(3):299-308
Let(X,θ),(X_1,θ_1),…,(X_n,θ_n)be iid.R~d×{1,2,…,s}-valued random vectors and letL_n be the posterior error probability in NN(nearest neighbor).diserimination.Someknowledge of the unknown value of L_n is of great meaning in many applications.For thisaim,in 1971,T.J.Wagner introduced an estimate of L_n which is defined by_n=1/nI(θ_j≠θ_(nj)),where θ_(nj) is the NN discrimination of θ_j based on the training samples(X_1,θ_1),…,(X_(j-1),θ_(j-1)),(X_(j+1),θ_(j+1)),…,(X_n,θ_n).Then he showed that _nR,where R is the limit ofthe prior error probability.But the problem of“)nR” is still left open since thattime.In this paper,it is shown that for any s>0,there exist two positive constants a andC such that P(丨_n-R丨≥ε)≤Ce~(-an).By this it is clear that _nR. 相似文献
9.
ON THE UNIQUENESS OF THE WEAK SOLUTIONS OF A QUASILINEAR HYPERBOLIC SYSTEM WITH A SINGULAR SOURCE TERM 下载免费PDF全文
This paper is a continuation of the authors'previous paper[1].In this paper the authorsprove,assuming additional conditions on the initial data,some results about the existence anduniqueness of the entropy weak solutions of the Cauchy problem for the singular hyperbolicsystem a_t+(au)_x_2au/x=0,u_t+1/2(a~2+u~2)_x=0,x>0,t≥0. 相似文献
10.
Cheng ping 《数学年刊B辑(英文版)》1984,5(3):357-362
Let X_1,…,X,be a sequence of p-dimensional iid.random vectors with a commondistribution F(x).Denote the kernel estimate of the probability density of F(if it exists)by_n(x)=n~(-1)h~_n(-p)K((x-X_i)/h_n)Suppose that there exists a measurable function g(x)and h_n>0,h_n→0 such thatlim sup丨f_n(x)-g(x)丨=0 a.s.Does F(x)have a uniformly continuous density function f(x)and f(x)=g(x)?This paperdeals with the problem and gives a sufficient and necessary condition for generalp-dimensional case. 相似文献
11.
A idempotent quasigroup (Q, o) of order n is equivalent to an n(n-1)×3 partial orthogonal array in which all of rows consist of 3 distinct elements. Let X be a (n+1)-set. Denote by T(n+1) the set of (n+1)n(n-1) ordered triples of X with the property that the 3 coordinates of each ordered triple are distinct. An overlarge set of idempotent quasigroups of order n is a partition of T(n+1) into n+1 n(n-1)×3 partial orthogonal arrays A_x, x∈X based on X\{x}. This article gives an almost complete solution of overlarge sets of idempotent quasigroups. 相似文献
12.
Bai zhengguo 《数学年刊B辑(英文版)》1988,9(1):32-37
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
13.
陈韵梅 《应用数学学报(英文版)》1985,2(3):191-212
In this paper,we discuss the problem for the nonlinear Schr(?)dinger equation(?)where Ω is the exterior domain of a compact set in B~n,a_j(u)=O(|u|),b_j(u)=O(|u|)(1≤j≤n),c(u)=O(|u|~2)near u=0.If n≥5,some Sobolev norm of u_0(x)is sufficiently small,under suitableassumptions on smoothnessand and compatibility and the shape of Ω,we get that the problem has aunique global solution u(t,x),with the decay estimate‖u(t,·)‖_(L(?)(Ω))=O(t~(-n/4)),‖u(t,·)‖_(L~4(Ω))=O(t~(-n/4)),t→+∞. 相似文献
14.
In this paper we give the exact order of丨x-x_k丨~丨l_k(x)丨~for any fixed nonnegativeintegers s and t,which is n~(-s),n~(-s)lnn and n~(1-)for s≤t-2,s=t-1 and s≥t,respectively. 相似文献
15.
In[1]we discussed the problems of calculus of variations withstrong nonlinearity.Its Euler equation is evidently the followingelliptic type equation of divergent form with strongly increasingcoefficients,(da(x,u,u_x))/(dx_) a(x,u,u_x)=0,x∈ΩR~n.(1)We get the interior regularities of the solutions of(1).It is justthe Hilbert's 19th problem in the ease with strong nonlinearity. 相似文献
16.
Lin Zhengsheng 《数学年刊B辑(英文版)》1984,5(3):363-373
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
17.
Zhang Hua Sheng Wancheng 《高校应用数学学报(英文版)》2006,21(1):30-40
§1IntroductionIn the present paper,we are interested in solving the Cauchy problem for linearizedsystem of two-dimensional isentropic flow with initial data in gas dynamicsρt+ρ0xu+yv=0,ut+p′(ρρ00)xρ=0,vt+p′(ρ0ρ0)yρ=0,(1.1)t=0:(ρ,u,v)=(ρ0(r),u0(r),v0(r)),(1.2)whereρis the density,(u,v)is the velocity,ρ0is a positive constant,p=p(ρ)is theequation of state satisfying p′(ρ0)>0,(r,θ)is the polar coordinate such thatx=rcosθ,y=rsinθ,0≤r<+∞,0≤θ≤2π… 相似文献
18.
This paper deals with the following IBV problem of nonlinear hyperbolic equations u_(tt)- sum from i, j=1 to n a_(jj)(u, Du)u_(x_ix_j)=b(u, Du), t>0, x∈Ω, u(O, x) =u~0(x), u_t(O, x) =u~1(v), x∈Ω, u(t, x)=O t>O, x∈()Ω,where Ωis the exterior domain of a compact set in R~n, and |a_(ij)(y)-δ_(ij)|= O(|y|~k), |b(y)|=O(|y|~(k+1)), near y=O. It is proved that under suitable assumptions on the smoothness,compatibility conditions and the shape of Ω, the above problem has a unique global smoothsolution for small initial data, in the case that k=1 add n≥7 or that k=2 and n≥4.Moreover, the solution ham some decay properties as t→ + ∞. 相似文献
19.
陈国旺 《数学物理学报(B辑英文版)》1991,(4)
In this paper, the periodic boundary problem and the initial value problem for the nonlinear system of parabolic type u_1=-A(x, t)u_(x4)+B(x, t)u_(x2)+(g(u))_(x2)+(grad h(u))_x+f(u)are studied, where u(x, t)=(u_1(x, t).…, u_J(x, t) is a J-dimensional unknown vector valued function, f(u) and g(u) are the J-dimensional vector valued function of u(x, t), h(u) is a scalar function of u, A(x, t) and B(x, t) are J×J matrices of functions. The existent, uniqueness and regularities of the generalized global solution and classical global solution of the problems are proved. When J=1, h(u)=0, g(u)=au~3, A=a_1, B=a_2, where a_1, a_2 a are constants, the system is a generalized diffusion model equation in population problem. 相似文献
20.
§1. Introduction In [1], for any α>0, and a function φ defined on [0,1], Geng-Zhe Change defined the generalized Bernstein-Bezier polynomial ofφ as follows: B_(n, a)(φ, x) = sum from k=0 to n φ(k/n){f_(nk)~a(x)-f_(n,k+1)~a,(x)} (1.1)where f_(n, n+1) (x) =0 and f_(n, k)(x) = sum from j=k to n x~j(1-x)~(n-j) k=0,1,...,n. (1.2)are the Bezier base functions of degree n.Obviously, for any x ∈(0, 1), we have 相似文献