常系数线性齐次递归式的一般解公式

本文给出常系数线性递归式 a_n=α_1a_(n-1)+α_2a_(n-2)+…+α_pa_(n-p),a_0=c_0,a_1=c_1,…,a_(p-1)=c_(p-1)的一般解公式 a_n=sum from k=0 to p-1(sum from i=k to p-1 c_iα_(p-i+k))F_(n-p-k)(n≥p),其中(?)     

GF(p)上B_n的次数

行列式 B_n=∑±b_(i_1)~(m_1)b_(i_2)~(m_2)…b_(i_n)~(m_n)中各项含因子 b 的个数的最大值称为 B_n 的次数,其中,1≤t_k≤n,m_f≥0,b_(i_k)∈GF(p).当 p=2时,这是0-1矩阵的行列式,文[3]已有结果.本文在任意 p 的情形下给出 B_n 的次数 L(n)的公式:对任意正整数 r,当 n_r≤n≤n_(r+1)时,L(n)=r,其中,n_r=(r_0+1)(p~(q+1)-1)/(p-1)-(1+qp~(q+1),q=[r/(p-1)],r=q(p-1)+r_0。     

B值鞅的强大数定律

本文证明了在 p 阶光滑空间中的 B 值鞅差序列{D_n},若存在 q≥1及递增的正实数列{a_n},a_n↑∝(n→∝),满足sum from n=1 to ∞ a_n~(-p)(a_n~p-a_(n-1)~p)~(-(q-1))E‖D_n‖~(pq)<∝ (a_0=0)则(sum from i=1 to n D_i)/a_n→0 a.s.(n→∝)并得到了 p 阶光滑空间特征的两个新刻划,应用到其它几种 B 值鞅型序列,也获得一些结果。     

二次指派问题的一个新的限界方法

二次指派问题(QAP)的数学模型是:min{z(x)=sum from i=1 to n sum from =1 to n a_(ip)x_(ip)+sum from i=1 to n sum from p=1 to n sum from j=1 to n sum from q=1 to n c_(ipjq)x_(ip)x_(jq)|x∈},(1)这里∈(n~2维布尔集)是满足如下约束的集合:sum from i=1 to n x_(ip)=1,1≤p≤n,(2)sum from p=1 to n x_(ip)=1,1≤i≤n,(3)x_(ip)=0,1,1≤i,p≤n.(4)因为 x_(ip)~2=x_(ip)并且有约束(2)和(3),我们可以约定 c_(ipjq)=0,当 i=j 或 p=q.如果所有二次项的系数都可以写成     

Hilbert重级数定理的一个改进

The object of this note is to prove the followingTheorem Let{a_n}and{b_n}be sequences of real numbers such that0<∑∑a_n~2<+∞and0<∑b_n~2<+∞.Then we have the inequalitysum from m=1 to∞sum from n=1 to∞a_mb_n/m+n<{sum from n=1 to∞(π-θ/n~(1/2)a_n~2}~1/2{sum from n=1 to∞(π-θ/n~(1/2)b_n~2}~1/2 (1)whereθ=3/2~(1/2)-1=1.121320343.     

一个关于非负矩阵的不等式

证明:若(xij)是一个元素不全为零的m×n非负矩阵,则当0相似文献   

Goldbach猜想的一种新尝试(英文)

设N为大偶数,以D(N)表示将N表成两个素数之和的表法个数,即 D(N)=sum from N=P_1+P_3 (1)。Hardy和Littlewood利用“圆法”证明了下面的结果 D(N)=(?)(N)N/log~2N+R (1)这里 (?)(N) 2 multiply from p>2((1-1/(p-1)~2) multiply from p\N P>2 (1+1/p-2),(2) R=(sum from q>Q(μ~2(q)/φ~2(q))C_q(-N))N/log~2N+integral from E (S~2(α,N)e~(-2πtαN)dα) (3) S(α,N)=sum from p≤N (e~(2πiαp)),C_q(-N)=sum from n=1 to q (e~(2πiNh/q))Q=log~(16)N,E表示在通常意义下的余区间,这就提出了下面的猜想 D(N)～(?)(N)N/log~2·(4)熟知Goldbach猜想的困难在于误差项R的处理,至今“圆法”是提出猜想(4)的唯一的方法,本文提出了另一种途径来研究猜想(4)。而且方法是初等的,看起来是更为直接的方法。令 (?)(N)=sum from d≤N(Λ(d)Λ(N-d))。 显然 D(N)=(?)(N)/log~2N[1+O(log log N/log N)]+O(N/log~3N).本文证明了下面两个定理: 定理1 设N为大偶数,这里证明定理1的方法是初等的,这就建议我们提出猜想(4)。 定理2 用Bombieri定理可以证明 R_1=R_2=O(Nlog~(-1)N)。从上面两个定理看出,研究Goldbach猜想的困难,在于处理余项R_3。     

New Inverse Series Relations for Finite and Infinite Series with Applications

Some new series inversion formulas of the general form F(n)=sum form k=0 to r(A_(k,m)f(n-mk)) if and only if f(n)=sum form k=0 to r(B_(k,n)F(r_o-mk)) valid for either r=[n/m]or r=∞ are presented. These relations generalize many of those given by the author in a long series of preceding papers. An interesting example is given by A_(k,n)=(-y)~kA_k(p-λn,t(1+λm)) and B_(k,n)=y~kA_k(p-λn,(1-t)(1+λm)) where A_k(a,b)=a/(a+bk) in terms of binomial coefficients. Here p,t,y and λ are arbitrary complex numbers. A corresponding Abel coefficient case occurs which uses numbers of the form a(a+bi)(i-1)/i!. An application to special functions studied by Singhal and Kumari is given, and it is also shown that sum form k=0 to ∞(z~kA_k(a+ck,b))=x~a(x-b(x-1))/(x-(b+c)(x-1)), where z=(x-1)x~(-b-c), with a corresponding case for the Abel coefficients sum from k=0 to ∞(z~kB_k(a+ck,b))=x~o(1-b logx)/(1-(b+c)log x),where z=(log x)x~(-b-c) From these expansions we then have easily the new convolution formula for Rothe coeffici     

High-dimensional D. H. Lehmer problem over short intervals

Letk be a positive integer and n a nonnegative integer,0 λ1,...,λk+1 ≤ 1 be real numbers and w =(λ1,λ2,...,λk+1).Let q ≥ max{[1/λi ]:1 ≤ i ≤ k + 1} be a positive integer,and a an integer coprime to q.Denote by N(a,k,w,q,n) the 2n-th moment of(b1··· bk c) with b1··· bk c ≡ a(mod q),1 ≤ bi≤λiq(i = 1,...,k),1 ≤ c ≤λk+1 q and 2(b1+ ··· + bk + c).We first use the properties of trigonometric sum and the estimates of n-dimensional Kloosterman sum to give an interesting asymptotic formula for N(a,k,w,q,n),which generalized the result of Zhang.Then we use the properties of character sum and the estimates of Dirichlet L-function to sharpen the result of N(a,k,w,q,n) in the case ofw =(1/2,1/2,...,1/2) and n = 0.In order to show our result is close to the best possible,the mean-square value of N(a,k,q) φk(q)/2k+2and the mean value weighted by the high-dimensional Cochrane sum are studied too.     

关于B值随机元赋范部分和最大值的矩的若干讨论

本文讨论B值随机元部分和序列的最大值的矩的问题,对1≤p≤2及r>p证明了下列叙述的等价性; (ⅰ)存在常数0相似文献   

非线性Neumann问题正解的存在性

研究非线性Neumann问题(p(t)u′)′+q(t)u=f(t,u),t∈(0,1),u′(0)=u′(1)=0正解的存在性,其中p,q∈C[0,1]满足p(t)>0,0*,t∈[0,1],b*,t∈[0,1],b^*为线性问题(p(t)u′)′+bu=0,u′(0)=0,u(1)=0的第一特征值.运用拓扑度理论及Rabinowitz全局分歧定理为上述问题建立了正解的存在性结果.     

关于Fujita型反应扩散方程组的Cauchy问题

本文研究Ｆｕｊｉｔａ型反应扩散方程组ｕｔ－Δｕ＝α１｜ｕ｜ｑ１－１ｕ＋β１｜ｖ｜ｐ１－１ｖ，（ｘ∈ＲＮ，ｔ＞０），ｖｔ－Δｖ＝α２｜ｕ｜ｑ２－１ｕ＋β２｜ｖ｜ｐ２－１ｖ，ｕ（ｘ，０）＝ｕ０（ｘ）０，ｖ（ｘ，０）＝ｖ０（ｘ）０，（ｘ∈ＲＮ）Ｌｐ解的整体存在性和有限时间Ｂｌｏｗ ｕｐ问题．这里ｑｉ＞１，ｐｉ＞１（ｉ＝１，２），α１０，α２＞０，β１＞０，β２０，１ｐ＋∞．     

Existence of solutions for nonlinear singular boundary value problems

Existence of solutions to the two-point boundary value problem (p(t)y')' = q(t)f(t, y,p(t)y'), y(l) = 0, lim_t→0+ p(t)y'(t) = 0 is established under a variety of conditions. Here p(0) = 0 is allowed, and q is not assumed to be continuous at 0, so the problem may be doubly singular. In addition, the Dirichlet problem for this differential equation is investigated     

一个带非线性边界条件的强耦合抛物方程组的整体存在性与爆破

本文处理带非线性边界条件 u n=uα, v n=vβ ,(x ,t) ∈ Ω× (0 ,T)的抛物方程组ut =vpΔu ,vt=uqΔv ,(x ,t) ∈Ω× (0 ,T) ,其中Ω RN 为一个有界区域 ,p ,q>0和α ,β≥ 0为常数 .研究了上述问题正解的整体存在性和爆破 ,建立了整体存在和爆破的新标准 .证明了当max{p+β,q+α}≤ 1时正解 (u ,v)整体存在 ,当min{p+β ,q+α}>1且max{α ,β}<1时正解 (u ,v)在有限时刻爆破     

二阶离散周期边值问题的多重正解

考虑如下边界值问题:-Δ[p(n-1)Δy(n-1)]+q(n)y(n)=f(n,y(n)),n∈[1,N](1.1)y(0)=y(N),p(0)Δy(0)=p(N)Δy(N)(1.2)其中{y(n)}nN=+01是一个期望解.运用锥不动点定理,给出了一种二阶离散周期边值问题多重正解的新的存在性定理.     

一维奇异p-Laplace方程的上下解方法[英文]

本文讨论了一维奇异 p Laplace方程( φp( y′) )′+ q(t) f(t,y) =0 ,0 相似文献   

一类具有非线性内部源和边界流的双重退化抛物型方程

This paper deals with blow-up criterion for a doubly degenerate parabolic equation of the form (un)t = (|ux|m-1ux)x up in (0, 1) × (0, T) subject to nonlinear boundary source (|ux|m-1ux)(1,t) = uq(1,t), (|ux|m-1ux)(0,t) = 0, and positive initial data u(x,0) = uo(x), where the parameters m, n, p, q ＞ 0.It is proved that the problem possesses global solutions if and only if p ≤ n and q≤min{n, m(n 1)/ m 1}.     

Dissipative Elliptic Problems in Domains with Cylindrical Ends, Scattering Matrices, and Radiation Conditions

In a domain with cylindrical ends at infinity, we consider a general elliptic dissipative boundary value problem. The coefficients of the imaginary part of the operator of the problem vanish as The asymptotic behavior of the solutions is expressed in terms of incoming and outgoing waves (the amplitudes of such waves can grow at infinity). We introduce an (augmented) scattering matrix and, in terms of this matrix, we compute the number of linearly independent solutions to the homogeneous problem vanishing at infinity with a given rate. We discuss the statement of a problem with the so-called radiation conditions. The natural radiation conditions (only outgoing waves occur in asymptotic formulas for solutions) can be applied in any case. Other admissible radiation conditions for the problem under consideration are connected with the natural ones via scattering matrices. Bibliography: 12 titles.     

An $L_1$ Minimization Problem by Generalized Rational Functions

Let $P,Q \subset L_1(X,\Sigma,\mu)$ and $q(x)>0$ a. e. in $X$ for all $q\in Q$. Define $R=\{p/q:p\in P,q\in Q\}$. In this paper we discuss an $L_1$ minimization problem of a nonnegative function $E(z,x)$, i.e. we wish to find a minimum of the functional $\phi(r)=\int _X qE(r,x)d\mu$ form $r=p/q\in R$. For such a problem we have established the complete characterizations of its minimum and of uniqueness of its minimum, when both $P,Q$ are arbitrary convex subsets.     

次线性奇异Dirichlet问题的正解

利用不动点指数理论,证明了奇异Dirichlet问题正解的存在性,其中函数q允许在t=0和t=1处奇异.有关结果可应用到非线性项可变号且下方无界的情形.