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1.
Suppose h∈L~2(R), α_0>1, b_0>0 and h_(mn) (x) =α_0~(-m/2)h(α_0~(-m)x - nb_0),m,n∈Zand suppose that {h_(mn)} is a frame with frame bounds A,B>0,where <·,·> is the standard inner product on L~2(R) and ||·|| is the L~2 norm on R .Wecall {h_(mn)} the affine frame. Denote its dual frame by {h_(mn)} .It is well known that forany f ∈L~2 (R), 相似文献
2.
ON SCORE VECTORS AND CONNECTIVITY OF TOURNAMENTS 总被引:1,自引:0,他引:1
姚天行 《数学年刊B辑(英文版)》1992,(2)
An n-tournament T is called k-strong (l≤k≤n-2), if every (n+1-k)-subtournament of T is strongly connected. This paper proves that a score vector (s_1, s_2,..., s_n), where s_1≤s_2≤...≤s_n, is the score vector of some k-strong tournament if and only if min{t_1, t_2,..., t_(n-1)}≥k, where t_f=s_1+s_2+...+s_j-j(j-1)/2, j=1, 2,..., n-1. 相似文献
3.
宋国柱 《数学物理学报(A辑)》1991,11(4):439-447
设X为Banach空间,T(t)为X上的(1,A)类半群,A为T(t)的无穷小母元,若对每个x∈X,映射t→T(t)x关于t>t_0可微,则称T(t)关于t>t_0可微,本文讨论了关于t>t_0可微的(1,A)类半群的若干性质,并利用可微半群母元豫解式的增长阶特征证明了关于t>t_0可微的(1,A)类半群是指数稳定的充分必要条件为sup{Reλ:λ∈σ(A)}<0. 相似文献
4.
<正> 设函数列{S_n(x)}在x_o点右方有定义,且■以及S(x_o+0)都存在.假如当n→∞,x→x_o时的二重上限大于S(x_o+0): lim sup S_n(x)>S(x_o+0),或者二重下限小于S(x_o+0): 相似文献
5.
《数学物理学报(B辑英文版)》2016,(2)
Necessary and sufficient conditions are studied that a bounded operator T_x =(x_1~*x, x_2~*x,···) on the space ?_∞, where x_n~*∈ ?_∞~*, is lower or upper semi-Fredholm; in particular, topological properties of the set {x_1~*, x_2~*,···} are investigated. Various estimates of the defect d(T) = codim R(T), where R(T) is the range of T, are given. The case of x_n~*= d_nx_(tn)~*,where dn ∈ R and x_(tn)~*≥ 0 are extreme points of the unit ball B_?_∞~*, that is, t_n ∈βN, is considered. In terms of the sequence {t_n}, the conditions of the closedness of the range R(T)are given and the value d(T) is calculated. For example, the condition {n:0 |d_n| δ} = Φ for some δ is sufficient and if for large n points tn are isolated elements of the sequence {t_n},then it is also necessary for the closedness of R(T)(t_(n0) is isolated if there is a neighborhood U of t_(n0) satisfying t_n ■ U for all n ≠ n0). If {n:|d_n| δ} =Φ, then d(T) is equal to the defect δ{_tn} of {t_n}. It is shown that if d(T) = ∞ and R(T) is closed, then there exists a sequence {A_n} of pairwise disjoint subsets of N satisfying χ_(A_n)■R(T). 相似文献
6.
7.
WANG SHENGWANG 《数学年刊B辑(英文版)》1980,1(34):325-334
1谱位于平面上的有界\[{\mathcal{D}_{ < {M_k} > }}\]型算子 记号与[1,2]相同,不再一一赘述.设序列
{Mk}满足(M.1),(M.2),(M.3)即.对数凸性、非拟解析性、可微性[1]. 由{M(k)}我们可以
定义二元相关函数\[M({t_1},{t_2})\](详见[7])以及二元\[{\mathcal{D}_{ < {M_k} > }}\]空间
\[{\mathcal{D}_{ < {M_k} > }} = \{ \varphi |\varphi \in \mathcal{D};\exists \nu ,st{\left\| \varphi \right\|_\nu } = \mathop {\sup }\limits_\begin{subarray}{l}
s \in {R^2} \\
{k_i} \geqslant 0 \\
(i = 1,2)
\end{subarray} |\frac{{{\partial ^{{k_1} + {k_2}}}}}{{{\partial ^{{k_1}}}{s_1}\partial _{{s_2}}^{{k_2}}}}\varphi (s)|/{\nu ^k}{M_k} < + \infty \} \]
其中\[s = ({s_1},{s_2})k = {k_1} + {k_2}\].关于谱位于复平面上的有界\[{\mathcal{D}_{ < {M_k} > }}\]型算子的定义及性质可
参看[3,4].设X为Banach空间,B(X)为X上有界线性算子的全体组成的环.当
\[T \in B(X)\]为\[{\mathcal{D}_{ < {M_k} > }}\]型算子时,有\[T = {T_1} + i{T_2};{T_1} = {U_{Ret}}{T_2}{\text{ = }}{U_{\operatorname{Im} {\kern 1pt} t}}\] ,此处U为T的谱超广义函数,t为复变量.由于supp(U)为紧集,故可将U延拓到\[{\varepsilon _{ < {M_k} > }}\]上且保持连续性.
经过简单的计算,若\[T \in B(X)\]为谱位于平面上的一个\[{\mathcal{D}_{ < {M_k} > }}\]型算子,则T的一个谱
超广义函数(1)U可表成
\[{U_\varphi } = \int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } {{e^{i({t_1}{T_1} + {t_2}{T_2})}}\hat \varphi } } ({t_1},{t_2})d{t_1}d{t_2}\]
设\[T \in B(X)\]为谱算子,S、N、E(.)分别为T的标量部分、根部、谱测度.下面的定理给出了谱算子成为\[{\mathcal{D}_{ < {M_k} > }}\]型算子的一个充分条件:
定理1设T为谱算子适合下面的条件
\[\mathop {\sup }\limits_{k > 0} \mathop {\sup }\limits_\begin{subarray}{l}
|{\mu _j}| < 1 \\
{\delta _j} \in \mathcal{B} \\
j = 1,2,...,k
\end{subarray} {(\left\| {\frac{{{N^n}}}{{n!}}\sum\limits_{j = 1}^k {{\mu _j}E({\delta _j})} } \right\|{M_n})^{\frac{1}{n}}} \to 0(n \to \infty )\]
其中\[\mathcal{B}\]为平面本的Borel集类.则T为\[{\mathcal{D}_{ < {M_k} > }}\]型算子且它的一个谱广义函数可表为
\[{U_\varphi } = \sum\limits_{n = 0}^\infty {\frac{{{N^n}}}{{n!}}} \int {{\partial ^n}} \varphi (s)dE(s)\]
推论1设E(?),N满足
\[{(\frac{{{M_n}}}{{n!}} \vee ({N^n}E))^{\frac{1}{n}}} \to 0\]
则T为\[{\mathcal{D}_{ < {M_k} > }}\]型算子.
推论2设N为广义幂零算子,则对于任何与N可换的标量算子S,S+N为\[{\mathcal{D}_{ < {M_k} > }}\]型算子的充分必要条件是
\[{(\frac{{\left\| {{N^n}} \right\|}}{{n!}}{M_n})^{\frac{1}{n}}} \to 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (n \to \infty )\]
在[4]中称满足上式的算子为\[\{ {M_k}\} \]广义幂零算子.显然\[\{ {M_k}\} \]广义幂零算子必为通
常的广义幂零算子.下面的命题给出了\[\{ {M_k}\} \] 广义幂零算子的一些性质.
命题 设N为广义幂零算子,则下列事实等价:
(i ) N为\[\{ {M_k}\} \]广义幂零算子;
(ii)对于任给的\[\lambda > 0\],存在\[{B_\lambda } > 0\]使(1)
\[\left\| {R(\xi ,N)} \right\| \leqslant {B_\lambda }{e^{{M^*}(\frac{\lambda }{{|\xi |}})}}\](\[{|\xi |}\]充分小);
(iii)对于任给的\[\mu > 0\],存在\[{A_\mu } > 0\]使
\[\left\| {{e^{izN}}} \right\| \leqslant {A_\mu }{e^{M(\mu |z|)}}\]
2谱位于实轴上的有界\[{\mathcal{D}_{ < {M_k} > }}\]型算子本节讨论有界\[{\mathcal{D}_{ < {M_k} > }}\]型算子T成为谱算子
的条件,这里假定\[{\mathcal{D}_{ < {M_k} > }}\]中的函数是一元的,于是Т的谱位于实轴上.X*表示X的共轭
空间.
设\[f \in {\mathcal{D}^'}_{ < {M_k} > }\],由[8, 9],存在测度\[{\mu _n}(n \geqslant 0)\]使得对任何h>0,存在A>0适合
\[\sum\limits_{n = 0}^\infty {\frac{{{h^n}}}{{n!}}} {M_n}\int {|d{\mu _n}| \leqslant A} \]且
\[ < f,\varphi > = \sum\limits_{n = 0}^\infty {\frac{1}{{n!}}} \int {{\varphi ^{(n)}}} (t)d{\mu _n}(t)\]
一般说,上述\[{\mu _n}(n \geqslant 0)\]不是唯一的,为此我们引入
定义设\[{n_0}\]为正整,如果对一切\[n \geqslant {n_0}\],存在测度\[{{\mu _n}}\],它们的支集均包含在某一L
零测度闭集内,则称f是\[{n_0}\]奇异的,若\[{n_0}\] = 1,则称f是奇异的.设\[T \in B(X)\]为\[{\mathcal{D}_{ < {M_k} > }}\]型
算子,U为其谱超广义函数,如果对于任何\[x \in X{x^*} \in {X^*},{x^*}U\].x是\[{n_0}\]奇异的(奇异
的),则称T是\[{n_0}\]奇异的(奇异的)\[{\mathcal{D}_{ < {M_k} > }}\]型算子.
经过若干准备,可以证明下面的
定理2 设X为自反的Banach空间,则\[T \in B(X)\]为奇异\[{\mathcal{D}_{ < {M_k} > }}\]型算子的充分必要
条件是T为满足下列条件的谱算子:
(i)对每个\[x \in X\]及\[{x^*} \in X\],\[\sup p({x^*}{N^n}E()x)\]包含在一个与\[n \geqslant 1\]无关的L零测
度闭集F内(F可以依赖于\[x{x^*}\]),此处E(?)、N分别是T的谱测度与根部;
(ii)算子N是\[\{ {M_k}\} \]广义幂零算子.
推论 设X为自反的banach空间,\[T \in B(X)\]为奇异\[{\mathcal{D}_{ < {M_k} > }}\]型算子且\[\sigma (T)\]的测度
为零的充分必要条件是T为满足下列条件的谱算子:
(i) E(?)的支集为L零测度集;
(ii) 算子N是\[\{ {M_k}\} \]广义幂零算子.; 相似文献
8.
Let μ be an Ahlfors-David probability measure on R~q;therefore,there exist some constants s_0 0 and ε_0,C_1,C_2 0 such that C_1ε~(s_0)≤μ(B(x,ε))≤C_2ε~(s_0) for all ε∈(0,ε_0) and x ∈ supp(μ).For n≥ 1,let α_n be an n-optimal set for μ of order r;furthermore,let {P_a(α_n)}_(a∈α_n) be an arbitrary Voronoi partition with respect to α_n.The n-th quantization error e_(n,r)(μ) for μ of order r can be defined as e_(n,r)~r(μ):=∫ d(x,α_n)~r dμ(x).We define I_a(α_n,μ):=∫_(P_a(α_n)) d(x,α_n)~r dμ(x),a ∈α_n,and prove that,the three quantities ■ are of the same order as that of 1/ne_(n,r)~r(μ).Thus,our result exhibits that,a weak version of Gersho's conjecture holds true for the Ahlfors-David probability measures on R~q. 相似文献
9.
分别记$T(\triangle)$与$B(\triangle)$为单位圆盘$\triangle$上的
Teichm$\mathrm{\ddot{u}}$ller空间与无限小Teichm$\mathrm{\ddot{u}}$ller空间.
证明了$[\nu]_{B(\triangle)}$是无限小Strebel点并不能说明$[\nu]_{T(\triangle)}$
是一个Strebel点以及$[\nu]_{T(\triangle)}$是Strebel点并不能说明$[\nu]_{B(\triangle)}$
是一个无限小Strebel点. 作为这个结论的应用,
解决了姚国武提出的问题. 相似文献
Teichm$\mathrm{\ddot{u}}$ller空间与无限小Teichm$\mathrm{\ddot{u}}$ller空间.
证明了$[\nu]_{B(\triangle)}$是无限小Strebel点并不能说明$[\nu]_{T(\triangle)}$
是一个Strebel点以及$[\nu]_{T(\triangle)}$是Strebel点并不能说明$[\nu]_{B(\triangle)}$
是一个无限小Strebel点. 作为这个结论的应用,
解决了姚国武提出的问题. 相似文献
10.
此处θ(x_1,y_1)表示分别由x_1与y_1所张成的两个1维子空间之间的夹角.Wilkinson指出,s_1~(-1)的大小反映了λ_1对于A的元素的变化的敏感性程度,因此s_1~(-1)被叫做单特征值λ_1的条件数. 现设λ_1是A的半单m_1重特征值(即λ_1的初等因子均为线性),?_1与?_1分别 相似文献
11.
设$X_1,X_2,\cdots,X_n$和$X^*_1,X^*_2,\cdots,X^*_n$分别服从正态分布$N(\mu_i,\sigma^2)$和$N(\mu^*_i,\sigma^2)$,以$X_{(1)}$,$X^*_{(1)}$分别表示$X_1,\cdots,X_n$和$X^*_1,\cdots,X^*_n$的极小次序统计量,以$X_{(n)}$, $X^*_{(n)}$分别表示$X_1,\cdots,X_n$和$X^*_1,\cdots$,$X^*_n$的极大次序统计量. 我们得到了如下结果:(i)\,如果存在严格单调函数$f$使得$(f(\mu_{1}),\cdots,f(\mu_{n}))\succeq_{\text{m}}$ $(f(\mu^{*}_{1}),\cdots,f(\mu^{*}_{n}))$,且$f'(x)f'(x)\!\geq\!0$, 则$X_{(1)}\!\leq_{\text{st}}\!X^*_{(1)}$;(ii)\,如果存在严格单调函数$f$使得$(f(\mu_{1})$,$\cdots,f(\mu_{n}))\succeq_{\text{m}}(f(\mu^{*}_{1}),\cdots,f(\mu^{*}_{n}))$,且$f'(x)f'(x)\leq 0$, 则$X_{(n)}\geq_{\text{st}}X^*_{(n)}$.(iii)\,设$X_{1},X_{2},\cdots,X_{n}$和\, $X^*_{1},X^*_{2},\cdots,X^*_{n}$分别服从正态分布$N(\mu,\sigma_i^2)$和$N(\mu,\sigma_i^{*2})$,若$({1}/{\sigma_{1}},\cdots,{1}/{\sigma_{n}})\succeq_{\text{m}}({1}/{\sigma^{*}_{1}},\cdots,{1}/{\sigma^{*}_{n}})$,则有$X_{(1)}\leq_{\text{st}}X^*_{(1)}$和$X_{(n)}\geq_{\text{st}}X^*_{(n)}$同时成立. 相似文献
12.
Let{T (t)}t≥0 be a C0-semigroup on an infinite-dimensional separable Hilbert space; a suitable definition of near{T (t)~*}t≥0 invariance of a subspace is presented in this paper. A series of prototypical examples for minimal nearly{S(t)~*}t≥0 invariant subspaces for the shift semigroup{S(t)}t≥0 on L2(0,∞) are demonstrated, which have close links with near T~*_θ invariance on Hardy spaces of the unit disk for an inner function θ. Especially, ... 相似文献
13.
在Banach空间X中,研究了如下半线性Caputo-分数阶中立型微分方程S-渐近w周期解的存在性其中0α1,-A是解析半群{T(t)}_(t≥0)的无穷小生成元. 相似文献
14.
设{u_k}_k≥0为一个线性递归序列.序列{u_k(mod q)}_(k≥0)是周期的,很多人都对其周期有过研究.本文应用二次数域中理想的理论,较完全地刻面了二次线性递归序列模q的周期长度,所获结果加强并推广了Engstrom及Wall的结论. 相似文献
15.
In this paper, the following retarded system has been studied
$\[\dot x(t) = Ax(t) + Bx(t - r),r > 0\]$(1)
where x(t) is an n-vector valued function; A and B are n*n constant matrices, and all the eigenvalues of A are supposed to have negative real parts. The asymptotical stability of equation (1) has been discussed by Halec13 utilizing the following Liapunov functional
$\[V(\phi ) = {\phi ^T}(0)C\phi (0) + \int_{ - r}^0 {{\phi ^T}(\theta )E\varphi (\theta )} d\theta \]$,
where E>0 and the symmetric matrix C>0 is chosen, such that A^TC+CA= — D<0. In this discussion, he remarked that if matrix
$\[H = \left[ {\begin{array}{*{20}{c}}
{D - E}&{ - CB}\{ - {{(CB)}^T}}&E
\end{array}} \right] > 0\]$,
the rate of decay of the solution of equation (1) to zero would be independent of the delay r, that is, would follow the exponential relation as indicated below : $\[||x(t,{t_0},\phi )|| \le K(r){e^{ - \alpha (t - {t_0})}}||\phi ||\]$,where \alpha(\alpha >0) is indepndent of r.
We show that this conclusion is not true, and a new relation between Liapunov functional and it's solution (exponential estimation) has been developed for the general rOtarded functional differential equation
$\[|\dot X(t) = f(t,{X_t})\]$(2)
If there is a functional $\[V(t,\phi ):{R^ + } \times {C_H} \to R\]$ such that
(i)$\[v|\phi (0){|^\eta } \le V(t,\phi ) \le K||\phi ||_\eta ^\eta ,(v,K > 0,\eta > 0)\]$
(ii)$\[\dot V(t,\phi ) \le - {C_1}|\phi (0){|^\eta },({C_1} > 0)\]$
then the solution of equation (2) x(t_0, ф) (t) satisfies
$\[||x({t_0},\phi )(t)|| \le {K_1}(r)||\phi |{|_\eta }{e^{ - {\alpha _1}(r)(t - {t_0})}}\]$
where \alpha _1 depends on r.
The following inverse problem has also been studied: In case the solution x = 0 of equation (1) is asymptotically stable for every value of r> 0, would there exist the matrices C>0 and E>0 such that the corresponding matrix H>0? Counter example is given for this problem. 相似文献
16.
Several Results on Systems of Residue Classes 总被引:2,自引:0,他引:2
Let (m,n) and a(n) denote the g.c.d, of m, n and the residue class {x∈Z∶x≡α (mod n)} respectively. Any period of the characteristic function ofkU a_i(n_i) is called a covering period of {a_i(n_i)}_(i-1)~k.i-ITheorem Let A = {a_i(n_i)}_(i-1)~k. be a disjoint system (i. e. a_I(n_I,...,a_k(n_k) are pairwise disjoint). Let [n_I,...,n_k] (the I.c.m. of n_1,...,n_k) have the prime faetorization [n_1,...,n_k] = Πp_i~ai and T = Πp_iβi(β_i≥0 be the smallest positive covering period of A. Then 相似文献
17.
18.
Li Xunjing 《数学年刊B辑(英文版)》1982,3(5):655-662
Let X and Z be two reflexive Banach spaces, U\in Z and b(\cdot,\cdot):[t_0,T]*U\rightarrow X continuous. Suppose $x(t)\equiv x(t,u(\cdot))$ is a function from [t_0, T] into X , satisfying the distrbnted parameter system
$dx(t)\dt=A(t)x(t)+b(t,u(t)),t_0+\int_t_0^T {
+r(t,u(t))dt}$. We have proved the following theorem. Theorem. Suppose u^*(\cdot) is the optimal control function, $x^*(t)=x(t,u^*(\cdot))$ and $\psi (t)=-U'(T,t)Q_1x^*(T)-\int_t^T{U'(\sigma,t)Q(\sigma)x^*(\sigma)d\sigma}$, then the maximum principle $<\psi(t),b(t,u^*(t))>-1/2r(t,u^*(t))=\mathop {\max }\limits_{u \in U} {\psi (t),b(t,u)>-1/2r(t,u)}$ (16) holds for almost all t on [t_0, T ]. 相似文献
19.
考虑了C1-正则半群{S(t)}t≥0和C2-正则半群{T(t)}t≥0之差Δ(t)=S(t)C1-T(t)C2在一定假定条件下的紧性. 相似文献
20.
考虑了具有耦合转移条件Sturm-Liouville(简称S-L)问题的逆问题,在一定条件下,通过利用S-L方程右边的函数f_j(x)确定方程的解,并由数据{u_j(x_0)}_j~∞=i或{p(x_0)(du_j(x_0))-(dx)}_j~∞=1唯一确定S-L算子中的系数p(x)和q(x).其中u_j(x)满足S-L方程,分离边界条件和耦合转移条件,而{f_j(x)}_(j-i)~∞构成L~2(I)的一个基. 相似文献