关于平稳序列中心秩顺序统计量联合分布的稳定收敛性

§1.引言 设{ξ_n}是平稳序列,ξ_1~(n)≤ξ_2~(n)≤…≤ξ_n~(n)是ξ_1,…,ξ_n的顺序统计量,则称{ξ_(kn)~(n)}{ξ_n}的具有秩序列{k_n}的顺序统计量序列。记λ_n=k_n/n和?_n={nλ_n·(1-λ_n}~(1/2),如果min{k_n,n-k_n}→∞或等价地?_n→∞就称{k_n}为变秩序列。     

非平稳高斯序列的极值之渐近分布

设{ξ_n}是一非平稳高斯序列,Eξ_n=0、Eξ_n~2=1及γ_(ij)=Eξ_iξ_j.以M_n记max ξ_k,以记公共分布是F(x)=/(2π)~(1/2) integral from n=-∞ to x(e~(-u~2/2))du的 i.i.d序列之前n个变量的最大值.已有如下结果:对所述非平稳高斯序列{ξ_n}若     

对称QL算法对角元的收敛条件

文[1]指出,在QL算法收敛性讨论中,仅有β_1~(K)→0并不能保证α_1~(k)收敛,并证明在加上条件:|α_1~(k)-σ_k|μ0”后,可确保α_1~(k)趋于T的某个固定特征值。本文首先对QL算法收敛性给出了一个精确的定义,然后给出一个与[1]不同的确保收敛的条件: “若{σ_k}_k=1~∞极限存在且β_i~(k)→0,则有α_i~(k)→λ_i(j=1,2,…,m)”条件“{σ_k}_k=1~∞极限存在”与“α_1~(k)-σ_k|→0”互不包含,在具体应用中,对后者无法判别(如[3]中给出的NS位移)或不成立的某些场合,前者具有独到的优点。     

WANDERING SUBSPACES OF THE HARDY-SOBOLEV SPACES OVER D~n

In this paper, we show that for log(2/3)/2log2≤β≤1/2, suppose S is an invariant subspace of the Hardy-Sobolev spaces H_β~2(D~n) for the n-tuple of multiplication operators(M_(z_1),...,M_(z_n)). If(M_(z_1)|S,..., M_(z_n)|S) is doubly commuting, then for any non-empty subset α = {α_1,..., α_k} of {1,..., n}, W_α~S is a generating wandering subspace for M_α|_S =(M_(z_(α_1))|_S,..., M_(z_(α_k))|_S), that is, [W_α~S]_(M_(α |S))= S, where W_α~S=■(S■z_(α_i)S).     

关于函数系{e~(-(?)_nx)x~(s_n-1)}的一些问题

在正实轴上考虑函数系,其中Reμ_n>0,n=1,2,…,且用S_k表示μ_k在{μ_1,μ_2,…,μ_k}中出现的次数,P_k表示μ_k在序列{U_n}_1~∞中出现的次数,已知 Mntz-Szasz定理:要使函数系在空间L~2[0, ∞)中完备,即对任意f(x)∈L~2《0, ∞),对任给ε>0,存在P_n(x)=sum from k=1 to n(c_ke~(-μ_ke~x)x~(S_(k-1)))使得     

SOME LIMIT PROPERTIES AND THE GENERALIZED AEP THEOREM FOR NONHOMOGENEOUS MARKOV CHAINS

Let(ξ_n)_(n=0)~∞ be a Markov chain with the state space X = {1, 2, ···, b},(g_n(x, y))_(n=1)~∞ be functions defined on X × X, and F_(m_n,b_n)(ω) =1 /b_n sum from k=m_n+1 to m_n+b_n g_k(ξ_(k-1), ξ_k).In this paper the limit properties of F_(m_n,b_n)(ω) and the generalized relative entropy density f_(m_n,b_n)(ω) =-(1/b_n) log p(ξ_(m_n,m_n+b_n)) are discussed, and some theorems on a.s. convergence for(ξ_n)_n=0~∞ and the generalized Shannon-McMillan(AEP) theorem on finite nonhomogeneous Markov chains are obtained.     

矩阵特征值的几个扰动定理

1 引言 设A∈C~(n×m),B∈C~(m×m)(m≤n),它们的特征值分别为{λ_k}_(k=1)~n和{μ_k}_(k=1)~m.令 R=AQ-QB (1)这里Q∈C~(n×m)为列满秩矩阵.Kahan研究了矩阵A在C~(n×m)上的Rayleigh商的性质,证明了下列定理:设A为Hermite矩阵,Q为列正交矩阵,即Q~HQ=I,而B=Q~HAQ,则存在 1,2,… ,n的某个排列π,使得 {sum from j=1 to m │μ_j-λ_(π(j))│~2}~(1/2)≤2~(1/2)‖R‖_F (2)其中R如(1)所示,‖·‖_F为矩阵的Frobenius范数.刘新国在[2]中将此定理推广到B为可对角化矩阵的情形,并且还建立了较为一般的扰动定理:设A为正规矩阵,B为可对角化矩阵;存在非奇异矩阵G,使得G~(-1)BG为对角阵,则存在1,2,…,n的某个排列π,使得 │μ_j-λ_(π(j))│≤2(2~(1/2))nK(G)_(σ_m~(-1))‖R‖_F,j=1,2,…,m. (3)     

THE COMPLEX FORM AND SOME BOUNDARY VALUE PROBLEMS FOR NONLINEAR ELLIPTIC SYSTEMS OF SECOND ORDER

The present paper is concerned with the nonlinear elliptic system of second order. Firstly, we shall establish a complex form of the system. Secondly .we shall consider the solvability of some boundary value problems for tbe complex equation of second order. let (1) \[{\Phi _j}(x,y,U,V,{U_x},{U_y},...,{U_{xx}},{U_{yy}},{V_{xx}},{V_{xy}},{V_{yy}}) = 0,j = 1,2\] be the I. G. Petrowkii’s nonlinear elliptic system of second Qrder in the botinded domain G, where \[{\Phi _j}(x,y,{z_1},...,{z_{12}})(j = 1,2)\]) are continuous real functions of the variables \[x,y[(x,y) \in G],{z_1},...,{z_{12}} \in R\], (the real axis), and contiriupusly differentiable for \[{z_1},...,{z_{12}} \in R\]. The solutions \[[U(x,y),V(x,y)]\], F(a?, y)] of the system are understood in the generalized sense. THEOBEM I. i) If the I. G. Petrovskii;s nonlinear system of equations (1) satisfies the M. I. visik-D. Xiagi’s uniformly elliptic condition for any solutions U(x,y),V(x,y) of (1) in G, then it can be written as the following complex equation? (2)\[{W_{z\overline z }} = F(z,W,{W_z},\overline {{W_z}} ,{W_{zz}},{\overline W _{zz}})\] where W=U+iV, z=x+iy, \[{W_z} = \frac{1}{2}[{W_x} - i{W_y}],...,\], ii) If the I. G. Petrovskii's nonlinear elliptic system (1) satisfies the condition that there exist two positive constants \[\delta \] and K, such that (3) \[|{\Phi _{j{U_{xx}}}}|,|{\Phi _{j{U_{xy}}}}|,|{\Phi _{j{U_{yy}}}}|,|{\Phi _{j{V_{xx}}}}|,|{\Phi _{j{V_{xy}}}}|,|{\Phi _{j{V_{yy}}}}| \leqslant K,j = 1,2\] \[|det(A)| \geqslant \delta > 0\], in G, then by a suitable linear trans-formation of the variables (x,y)into variables \[(\xi ,\eta )\], system (1) can be written as the following coinplex equation ⑷ \[{W_{\xi \xi }} = F(\xi ,W,{W_\xi },{\overline W _\xi },{W_{\xi \xi }},{\overline W _{\xi \xi }}),\varsigma = \xi + i\eta \] In the following section, we discuss the complex equation (2) of the following form: ,We^B(z9 Wee)x .\[(5)\left\{ \begin{gathered} {W_{zz}} = F(z,W,{W_z},{\overline W _z},{W_{zz}},{\overline W _{zz}}) \hfill \ F = {Q_1}{W_{zz}} + {Q_2}\overline {{W_{\overline z \overline z }}} + {Q_4}{W_{zz}} + {A_1}{W_z} + {A_2}{\overline W _{\overline z }} \hfill \ + {A_3}\overline {{W_z}} + {A_4}{W_{\bar z}} + {A_5}W + {A_6}\bar W + {A_7}, \hfill \ {Q_j} = {Q_j}(z,W,{W_{\bar z}},{\overline W _{\bar z}},{W_{zz}},{\overline W _{zz}}),j = 1,...,4 \hfill \ {A_j} = {A_j}(z,W,{W_z},{\overline W _z}),j = 1,...,7 \hfill \\ \end{gathered} \right.\] 1) \[{Q_j}(z,W,{W_z},{\overline W _z},U,V),j = 1,...,4.{A_j} = (z,W,{W_z},{\overline W _z}),j = 1,...,7\] are measurable functions of z for any continuously differentiable functions W(z) and measurable functions U(z), V(z) in G, Furthermore they satisfy (6)\[{\left\| {{A_j}} \right\|_{{L_p}(\overline {G)} }} \leqslant {K_0},j = 1,2,{\left\| {{A_j}} \right\|_{{L_p}(\overline {G)} }} \leqslant {K_1},j = 3,...,7\] where\[{K_0},{K_1}( \leqslant {K_0}),p( > 2)\] are constants: 2) Qj, Aj are continuous for \[W,{W_z},{\overline W _z} \in E\](the whole plane) and the continuity is uniform with respect to almost every point \[z \in G\] and \[U,V \in E\] 3) \[F(z,W,{W_z},{\overline W _z},U,V)\] satisfies the following Lipschitz's condition, i.e. for almost every point \[z \in G\], and for all \[W,{W_z},{\overline W _z}{U_1},{U_2},{V_1},{V_2} \in E\], the inequality (7)\[\begin{gathered} |F(z,W,{W_z},{\overline W _z},{U_1},{V_1}) - F(z,W,{W_z},{\overline W _z},{U_2},{V_2})| \hfill \ \leqslant {q_0}|{U_1} - {U_2}| + q_0^'|{V_1} - {V_2}|,{q_0} + q_0^' < 1 \hfill \\ \end{gathered} \] holds where \[{q_0},q_0^'\] are two nonnegative constants. In this paper, let G be a simply connected domain with boundary \[\Gamma \in C_\mu ^2(0 < \mu < 1)\]； without loss of geaerality, we may assume that G is the unit disk |z|<1. Now we, describe the results of the solvability of Riemann-Hilbert botindary value problem (Problem R-H) and the oblique derivative problem (Problem P) for Eq. (5) in the unit disk G: |z| <1. Problem R-H. We try to find a solution W（z）of Eq. (5) which is continuonsly differentiable on \[G\], and satisfies the boundary conditions: (8) \[\operatorname{Re} [{{\bar z}^{{\chi _1}}},{W_z}] = {r_1}(z),Re[{{\bar z}^{{\chi _2}}}\overline {W(z)} ] = {r_2}(z),z \in \Gamma \]? where \[{\chi _1},{\chi _2}\] are two integers, and \[{r_j} \in C_v^{j - 1}(\Gamma ),j = 1,2,\frac{1}{2} < v < 1\] Problem P. we try to find a solution W（z) of Eq. (5) which is continuously diffierentiabfe on \[\overline G \] and satisfies the boundaory conditions： (9) \[\operatorname{Re} [{{\bar z}^{{\chi _1}}}{W_z}] = {r_1}(z),Re[{{\bar z}^{{\chi _2}}}\overline {W(z)} ] = {r_2}(z),z \in \Gamma \], Where \[{\chi _1},{\chi _2},{r_1}(z),{r_2}(z)\] are the same as in (8), but \[{r_2}(z) \in {C_v}(\Gamma )\]. Theorem II. Suppose that Eq. (5) satisfies the condition C and the constants \[q_0^'\] and K1 are adequately small; then the solvability of Problem R-H is as follows： 1) When \[{\chi _1} \geqslant 0,{\chi _2} \geqslant 0\] Problem R-H is solvable; 2) When \[{\chi _1} < 0,{\chi _2} \geqslant 0(or{\kern 1pt} {\kern 1pt} {\chi _1} \geqslant 0,{\chi _2} < 0){\kern 1pt} \] there are \[2|{\chi _1}| - 1(or2|{\chi _2}| - 1)\] solvable conditions for Problem R-H; 3) WHen \[{\chi _1} < 0,{\chi _2} < 0\], there are \[2(|{\chi _1}| + |{\chi _2}| - 1)\] solvable conditions for Problem R-H. Theorem III Let Eq (5) satisf the condition C and the constants \[q_0^'\] and \[{K_1}\] are adequately small, then tbe solvability of Problem P is as follows： 1) When \[{\chi _1} \geqslant 0,{\chi _2} \geqslant 0\] Problem P is solvable; 2) When \[{\chi _1} < 0,{\chi _2} \geqslant 0(or{\kern 1pt} {\kern 1pt} {\chi _1} \geqslant 0,{\chi _2} < 0){\kern 1pt} {\kern 1pt} {\kern 1pt} \], there are \[2|{\chi _1}| - 1(or2|{\chi _2}| - 1)\] solvable conditions for Problem P; 3) When \[{\chi _1} < 0,{\chi _2} < 0\]; there are \[2|{\chi _1}|{\text{ + }}|{\chi _2}| - 1)\] solvable conditions for Problem P. Furthermore, the solution W(z) of Problem P for Eq. (5) may be expressed as \[{g_j}(\xi ,z) = \left\{ \begin{gathered} \int_0^z {\frac{{{z^{2{\chi _j} + 1}}}}{{1 - \bar \xi z}}dz,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} for{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\chi _j} \geqslant 0} \hfill \ \int_0^z {\frac{{{\xi ^{ - 2{\chi _j} - 1}}}}{{1 - \bar \xi z}}dz,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} for{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\chi _j} < 0} \hfill \\ \end{gathered} \right.j = 1,2\] where \[{\Phi _0}(z) = a + ib\] is a complex constant,and \[{\Phi _1}(z),{\Phi _2}(z)\] are two analytic functions. The proofs of the above stated theorems are based on a prior estimates for the bounded solutes of these boundary value problems and Leray-Schander theorem. Besides, we have considered also the solvability of Problem R-H and Problem P for Eq. (6) in the multiply connected domain.     

二阶线性系统稳定性判定的数值方法

A为N阶方阵,N=2n,设A的特征值为ξ_1,ξ_2,…,ξ_N,根据[3]有如下定义: 定义1.若所有的 Reξ_i<0,则称系统(1.1)是渐近稳定的,相应的A阵叫渐近稳定阵;若所有的Reξ_i≤0且对所有的Reξ_k=0的ξ_k对应的初级因子均是线性的,则称系     

几何测度空间基的研究

研究了几何测度空间中的基本对称函数μ_0,μ_1,…,μ_n和内蕴体积函数V_0,V_1,…,V_n,证明了Lⁿ上连续不变赋值函数空间中由基本对称函数构成的基{μ_0,μ_1,…,μ_n}和由内蕴体积函数构成的基{V_0,V_1,…V_n}(或均质积分构成的基{W_0,W_1,…,W_n})等价.     

随机元序列的收敛性

Slutsky 曾经证明下述定理:设随机变数序列(?)分别依概率收敛于常数 α_1,α_2,…,α_k,即 (?),i=1,2…,k 对任一给定的 ε>0成立,则对任一有理函数 R(x_1,x_2,…,x_k)当 R(α_1,α_2,…,α_k)有意义时必有 R(ξ(1n)ξ(2n)…,ξ(kn))依概率收敛于 R(α_1,α_2,…,α_k)。文献[1]推广了上述结果证明了 R 为 R~t(k 维欧氏空间)上的 Borel 函数,并在(α_1,α_2,…,α_k)处连续的条件下 Slutsky 定理仍成立。上述定理及其     

在Banach空间中推广的Roper-Suffridge算子(Ⅲ)

假定X是具有范数‖·‖的复Banach空间,n是一个满足dim X≥n≥2的正整数.本文考虑由下式定义的推广的Roper-Suffridge算子Φ_(n,β_22γ_2,…,β_(n+1),γ_(n+1))(f):(?)其中x∈Ω_(p1,p2,…,pn+1),β_1=1,γ_1=0和(?)这里p_j1(j=1,2,…,n+1),线性无关族{x_1,x_2,…,x_n}(?)X与{x_1~*,x_2~*,…,x_n~*}(?) X~*满足x_j~*(x_j)=‖x_j‖=1(j=1,2,…,n)和x_j~*(x_k)=0(j≠k),我们选取幂函数的单值分支满足(f(ξ)/ξ)~(β_j)|ξ=0=1和(f′(ξ))~(γ_j)|ξ=0=1,j=2,…,n+1.本文将证明:对某些合适的常数β_j,γ_j,算子Φ_(n,β_2,γ_2,…,β_(n+1),γ_(n+1))(f)在Ω_(p_1,p_2,…,p_(n+1))上保持α阶的殆β型螺形映照和α阶的β型螺形映照.     

关于极大强单调算子的不精确邻近点算法的收敛性分析

该文研究集值映象方程0∈T(z)的解的迭代逼近，其中T是极大强单调算子．设{x^k}与{e^k}是由不精确邻近点算法x^{k+1}+c_kT(x^{k+1})> x^k+e^{k+1}生成的序列，满足‖e^{k+1}‖≤η_k‖x^{k+1}_x^k‖, ∑^∞_{k=0}(η_k-1)<+∞且inf_(k≥0) η_k=μ≥1.在适当的限制下证明了，{x^k}收敛到T的一个根当且仅当 lim inf_{k→+∞} d(x^k,Z)=0，其中Z是方程0∈T(z)的解集     

UNBOUNDED SELF-ADJOINT OPERATOR ΠK SPACE

如果A是Πsubsub空间上的自共轭算子，由文[1]可知存在空间昨一个标准分解 \[{\Pi _k} = N \oplus \{ Z + {Z^*}\} \oplus P\] 在此分解下，A有三角模型\[A = \{ S,{A_N},{A_p},F,G,Q\} \].利用三角模型，我们直接证明了 定理1设A是\[{\Pi _k}\]上的-共轭算子，n是任何自然数，那末\[{A^n}\]也是自共轭算子. 定理2设A是\[{A^n}\]上的自共轭算子，那末对所有的\[{A^n}(n = 1,2,...)\]，存在一个公共 的标准分解，在此分解下 \[\begin{gathered} {A^n} = \{ {S^n},A_N^n,A_P^n,\sum\limits_{i = 0}^{n - 1} {{S^i}} FA_N^{n - 1 - i},\sum\limits_{i = 0}^{n - 1} {{S^i}GA_P^{n - 1 - i}} , \hfill \ \sum\limits_{i = 0}^{n - 1} {{S^i}} Q{S^{*n - 1 - i}} - \sum\limits_{i + j + k = n - 2} {{S^i}(FA_N^j{F^*} + GA_P^j{G^*}){S^{*k}}} \} \hfill \\ \end{gathered} \] 定理3 设A是瓜空间上的自共轭算子，\[\sigma (A) \subset [0,\infty ),0 \notin {\sigma _P}(A),\]，那末存在唯 一的自共轭算子A1，满足\[A_1^n = A,\sigma ({A_1}) \subset [0,\infty )\] 其次，我们研究了谱系在临界点附近的性状.记临界点全体为\[C(A)\]).对 \[{\lambda _0} \in C(A)\]记S与入0相应的最高阶根向量的阶数为\[r({\lambda _0})\] 定理4设A是\[{\Pi _k}\]空间上的无界自共轭算子，\[C(A) \cap ({\mu _1},{\nu _1}) = \{ {\lambda _0}\} \],那末以下四 个命题等价： (i）\[\mathop {sup}\limits_{\mu ,\nu } \{ \left\| {{E_{\mu \nu }}} \right\||{\lambda _0} \in (\mu ,\nu ) \subset ({\mu _1},{\nu _1})\} < \infty \] (ii）\[{\mu ^{{\text{1}}}}...,{\mu ^{{{\text{k}}_{\text{0}}}}}\]是全有限的测度； （iii）\[s - \lim {\kern 1pt} {\kern 1pt} {\kern 1pt} {E_{\mu \nu }}\]存在； （iv)A与\[{\lambda _0}\]相应的根子空间\[{\Phi _{{\lambda _0}}}\]非退化；这里\[{\mu ^{{\text{1}}}}...,{\mu ^{{{\text{k}}_{\text{0}}}}}\]是由\[{A_P}\]与G导出的测度. 定通5 设A是\[{\Pi _k}\]上自共轭算子，\[{\lambda _0} \in C(A),r({\lambda _0}) = n\]，那么 （i）\[{E_{\mu \nu }}\]在\[{{\lambda _0}}\]处的奇性次数不超过2n， （ii）\[s - \mathop {\lim }\limits_{\varepsilon \to 0} \int_{[{M_1},{\lambda _0} - \varepsilon )} {(t - {\lambda _0}} {)^{2n}}d{E_t},s - \mathop {\lim }\limits_{\varepsilon \to 0} \int_{[{\lambda _0} + \varepsilon ,{M_2})} {(t - {\lambda _0}} {)^{2n}}d{E_t},\]存在。这里\[{M_1},{M_2}\]满足\[[{M_1},{M_2}] \cap C(A) = \{ {\lambda _0}\} \] 定理6 设A是\[{\Pi _k}\]上的自共轭算子，临界点集\[C(A) = \{ {\lambda _1},...,{\lambda _l},{\lambda _{l + 1}},{\overline \lambda _{l + 1}},...,{\lambda _{l + p}},{\overline \lambda _{l + p}},\]，这里\[\operatorname{Im} {\lambda _v} = 0(1 \leqslant \nu \leqslant l),r({\lambda _\nu }) = {n_\nu }\]那么有 \[{(\lambda - A)^{ - 1}} = \int_{ - \infty }^\infty {K(\lambda ,t)d{E_t}} + \sum\limits_{\nu = 1}^l {\sum\limits_{i = 1}^{2{n_\nu } + 1} {\frac{{{B_{\nu i}}}}{{{{(\lambda - {\lambda _\nu })}^i}}}} } + \sum\limits_{\nu = l + 1}^{l + p} {\sum\limits_{i = 1}^{{n_\nu }} {[\frac{{{B_{\nu i}}}}{{{{(\lambda - {\lambda _\nu })}^i}}}} } + \frac{{B_{\nu i}^ + }}{{{{(\lambda - {{\overline \lambda }_v})}^i}}}]\] 这里 \[K(\lambda ,t) = \frac{1}{{\lambda - t}} - \sum\limits_{v = 1}^l {\delta (t - {\lambda _v}} )\sum\limits_{i = 1}^{2{n_v}} {\frac{{{{(t - {\lambda _v})}^{i - 1}}}}{{{{(\lambda - {\lambda _v})}^i}}}} ,\delta \lambda {\text{ = }}\left\{ \begin{gathered} {\text{1}}{\text{|}}\lambda {\text{| < }}\delta \hfill \ {\text{0}}{\text{|}}\lambda {\text{|}} \geqslant \delta \hfill \\ \end{gathered} \right.\] \[0 < \delta < \mathop {\min }\limits_\begin{subarray}{l} 1 \leqslant \mu ,v \leqslant l \\ {\lambda _\mu } \ne {\lambda _v} \end{subarray} |{\lambda _\mu } - {\lambda _v}|\].对\[1 \leqslant v \leqslant l\]，\[{B_{vi}}\]是\[{\Pi _k}\]上的有界自共轭算子，而当\[l + 1 \leqslant v \leqslant l + p\]时，\[{B_{vi}} = {({\lambda _\mu } - S)^{i - 1}}{P_{\lambda v}}\]是以与\[{{\lambda _v}}\]相应的根子空间为值域的某些平行投影. 定理7 在定理6的条件下，有 \[\begin{gathered} {\text{f}}(A) = \int_{ - 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