具有正密度部分商序列的连分数 [英文]

u_{x} = v_{x}F(u), u_{y} = v_{y}F(u), u_{z}=v_{z}F(u)\right\}$不变. 通过取特殊的$v$, 得到一些特殊的波方程在伸缩群、旋转群以及推广的伸缩和旋转群下不变的精确解,~并将该方法推广到(N+1) 维波方程的情形.     

The Estimation for the Multiple de la ValKe Poussin Square Remainders

Let Q_N={\bar x=(x_1,\cdots ,x_N)|-pi \leq x_i <\pi,i=1,\cdots,N} and X(Q_N) denote L(Q_N) and C(Q_N) , The square de la УаДбо Poussin sums of f\in X (Q_N) are defined by $V_n^n+l(f;\bar x)=\frac{1}{\pi ^N}\int _Q_N f(\bar x+\bar t)\prod\limits_{i = 1}^N {(\frac{1}{{l + 1}}} \sum\limits_{v = n}^{n + l} {{D_v}({t_i}))d\bar t(n,l = 0,1,2, \cdots )}$ where D_v(t) =sin(v+1/2)t/2sint/2, - The differences $R_n,l(f;\bar x)=f(\bar x)-V_n^n+l(f;\bar x)$ are called square remainders. We denote by E_k(f)_X the best approximation of the function f\in X(Q_N) by N-multiple trigonometric polynomials of order K. Theorem Let {\varepsilon _k}_k=0^\infty be a sequence such that \varepsilon _n \downarrow \infty(n\rightarrow \infty), the class $X(\varepsilon)={f\in X(Q_N)|E_k(f)_X \leq \varepsilon _k,k=0,1,2,\cdots}$ Then $C_N^'\sum\limits_{v=0}^n+l \frac {\varepsilon_v+nln^N-1(3+v/(l+1))}{v+l+1}\leq sup_{f\in X(\varepsilon)||R_n,l(f)||_X\leq C_N \sum\limits_{v=0}^{n+l}\frac {\varepsilon _v+nln^N-1(3+v/l+1)}{v+l+1}$ where C_N>C'_N>0 are constants depending only on N.     

齐次~Morrey-Herz 空间上分数次积分算子高阶交换子的有界性

在齐次Morrey-Herz空间上建立了高阶交换子~$T^{m}_{b,l}$ 和 ~$M^{m}_{b,l}$的有界性，其中~$T^{m}_{b,l}$ 和 ~$M^{m}_{b,l}$ 是由分数次积分算子和分数次极大算子分别与~BMO($R^{n}$)函数生成的高阶交换子.     

名校基础知识自测

★高一年级 北京第十二中学(100071)李有毅一、选择题1.卜列四个关系式中正确的是(). (A)g任{a}(B)a星{a} ((、){a}任{a,b}(D)a〔{a,b}2.满足{l}里A里{1,2,3}的集合A的个数为().(A)l(B)2(C)3(D)43.已知尸一{、}二2一3二+2一0},T一{y{yS一定之一一5}.则尸nTUS一().(A)2)(B){1,2}(C){一2,2}(D){1}设全集u一{2,3,5},A={}a一5{,2},CoA一{5},贝日u的值为().(A)2(B)8(C)2或8(D)一2或8已知集合{‘·{一2了.>2了,·>。)一{工}了<一5或二>4},则,丫+n的值为().(A)一8(11)l()(C)8(D)80若集合A一{二i“厂一a二+1<。}一②,则实数“的值的集合…     

SOME SIMPLE GROUPS OF LIE TYPE CONSTRUCTED BY THE INNER-AUTOMORPHISM

当\[L \cong {C_l}\]，l为偶数，且l≥4,域\[\mathcal{H}\]=\[{\mathcal{H}_0}(\sqrt { - 1} )\]，其中\[{\mathcal{H}_0}\]为一有序域（或\[{\mathcal{H}_0}\]满足： а)\[\sqrt { - 1} \notin {\mathcal{H}_0},{(\sqrt { - 1} )^2} = - 1\]; b)\[ch{\mathcal{H}_0} > 3\]; c)若\[a,b \in {\mathcal{H}_0}\]，则 \[{a^2} + {b^2} \ne - 1\]，设Ф和 \[\prod :\{ {\alpha _1},{\alpha _2},...,{\alpha _l}\} \]，\[{\alpha _l}\]为长根分别为L的一组根系和素根系.令\[\{ {h_r},r \in \prod ,{e_r}r \in \Phi \} \]为L的一组Chevalley基;\[G = L({\cal H})\]为对于这一组Chevalley基在域\[{\cal H}\]上的L型 Chevalley群,令\[{w_0} = {w_{{\alpha _1}}}{w_{{\alpha _2}}}...{w_{{\alpha _{l - 1}}}}\]，其中\[{\alpha _i} \in \prod \]且为对于垂直于\[{\alpha _i}\]的平面的反射，显然\[{w_0}\]为L的Weyl群中的元素.设N为G的单项子群，\[{n_0} \in N\]，\[{n_0}\]的自然同态 像为 \[{w_0}\]，且\[{n_0}^2{\rm{ = }}I\]，存在域\[{\cal H}\]的自同构f:f(a)=a,\[a \in {{\cal H}_0}\] , \[{\rm{f(}}\sqrt { - 1} {\rm{) = }} - \sqrt { - 1} \],f在G中的扩充为G的一个域自同构(仍记为f)，且令U(V)为G对于正(负)根生 成的么幂子群，令\[{U^1}\{ u \in U|{n_0}f(u){n_0}^{ - 1} = u\} \];\[{V^1}\{ v \in V|{n_0}f(v){n_0}^{ - 1} = v\} \], 本文证明了 \[{}^2{C_l}({\cal H}) = < {U^1},{V^1} > \]为一单群.     

ON DISCRETE PHENOMENA IN THE MIXED PROBLEM

On discrete phenomena in uniqueness of the initial value problem, F. Treves studied an interesting example and proved that the Oauohy problem \[\left\{ \begin{array}{l} {L_p}u = {u_{xx}} - {x^2}{u_{tt}} + p{u_t} = 0,t \ge 0;\u(x,0) = {u_t}(x,0) = 0, \end{array} \right.\] has non-triyial solutions if and only if p = 3, 5, …. Wang Guang-ymg and others proved that the Oauohy problem \[\left\{ \begin{array}{l} {L_p}u = 0,t \ge 0;\u(x,0) = {\varphi _1}(x);{u_t}(x,0) = {\varphi _2}(x), \end{array} \right.\] and Goursat problem \[\left\{ \begin{array}{l} {L_p}u = 0,t \ge \frac{{{x^2}}}{2};\u(x,\frac{{{x^2}}}{2}) = {\varphi _3}(x), \end{array} \right.\] both have a unique solution if and only if p≠1, 3, 5, …. In this paper, we discuss in detail the equation Lvu = 0 for discrete phenomena. We prove that solution of the mixed problem \[\left\{ \begin{array}{l} {L_p}u = 0,x \ge 0,t \ge 0,\u(x,0) = \varphi (x),\{u_t}(x,0) = \psi (x),\u(0,t) = 0 \end{array} \right.\] is not only existent but also unique, for р≠3, 7, 11，…，neither existence nor uniqueness could be proved in this problem, for p = 3, 7, 11，….，more precisely, only under some compatibility condition can the solution exist for the equation \({L_p}u = 0\).     

六、数列、极限、数学归纳法

A组一、选择题(有卜t仅有个答案正确)1.已知数列{凡}中,N=1,人,! 人,二I+止,则数列讯}一定是((A)丫又为等差数列,(B)f又为等比数列; ((、既j卜等差、又非等比数列;(D)既是等差、又是等比数列。2.卜列四个定义在自然数集上的函数: (l)f(。)=2,厂l, (2)g(,:)=一r,‘+6,,了一9,,+5 (3)h(r,)=,,‘一6,z兰+11,,一7, (4)k(,I)=,:‘已们可以是数列l, 6一q尹I十1仁J一公 r已 (B)在区l’ed(l一￡,l+〔)内,存在{a。}的无穷多项; ‘〔今在区I’ti](l一,l+￡)内,存在{a.}的有穷多项; (D)在(l一,1+￡)内和外,均存在{。}的无限多项. 8.在△ABC中,若…     

一类非线性分数阶薛定谔-泊松系统非零解的存在性

本文研究了分数阶薛定谔-泊松系统$$\left\{\begin{array}{l}(-\Delta)^su+u+\phi u=\lambda f(u)\ \text {in} \ \mathbb {R}^3, \\ (-\Delta)^{\alpha}\phi =u^2\ \text {in} \ \mathbb {R}^3\emph{},\end{array}\right. $$ 非零解的存在性, 其中$s\in (\frac{3}{4},1), \alpha\in(0,1),\lambda$ 是正参数, $(-\Delta)^s,(-\Delta)^{\alpha}$是分数阶拉普拉斯算子. 在一定的假设条件下, 利用扰动法和Morse迭代法, 得到了系统至少一个非平凡解.     

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 \[\begin{gathered} {\text{f}}(A) = \int_{ - \infty }^\infty {[f(t) - \sum\limits_{v = 1}^l {\delta (t - {\lambda _v}} } )\sum\limits_{i = 0}^{2{n_v} - 1} {\frac{{{f^{(i)}}({\lambda _v})}}{{i!}}} (t - {\lambda _v})d{E_t} \hfill \ {\text{ + }}\sum\limits_{{\text{v = 1}}}^{\text{l}} {\sum\limits_{i = 0}^{2{n_v}} {\frac{{{f^{(i)}}({\lambda _0})}}{{i!}}} } {B_v} + \sum\limits_{v = l + 1}^{l + p} {\sum\limits_{i = 0}^{{n_v} - 1} {[\frac{{{f^{(i)}}({\lambda _v})}}{{i!}}} } {B_{vi}} + \frac{{{f^{(i)}}({{\overline \lambda }_v})}}{{i!}}B_{vi}^ + ] \hfill \\ \end{gathered} \] 这里\[f(\lambda )\]在\[\sigma (A)\]的一个邻域内解析. 为了建立更一般的算子演算，我们引入两个特殊的代数： \[{\Omega _n} = \{ (f,\{ {a_i}\} _{i = 0}^{2n})|f\]为Borel可测函数，\[\{ {a_i}\} \]为一常数}。对\[F = (f,\{ {a_i}\} ) \in {\Omega _n},G = (g,\{ {b_i}\} ) \in {\Omega _n}\]，定义 \[\begin{gathered} \alpha F + \beta G = (\alpha f + \beta G,\{ \alpha {a_i} + \beta {b_i}\} ) \hfill \ F \cdot G = (f \cdot g,\{ \sum\limits_{j = 0}^i {{a_j}} {b_{i - j}}\} ),\overline F = (\overline f ,\{ {\overline a _i}\} ) \hfill \\ \end{gathered} \] 显然\[{\Omega _n}\]是一个交换代数，它的子代数\[{\omega _n}\]定义为 \[{\omega _n} = \{ F = (f,\{ {a_i}\} ) \in {\Omega _n}|\]在0点的一个与F有关的邻域中，成立\[{\text{|f(t) - }}\sum\limits_{i = 0}^{2n} {a{t^i}} | \leqslant {M_F}|t{|^{2n + 1}},{M_F}\]与F有关} 定义 设A是\[{\Pi _k}\]上的自共轭算子，C（A）={0}，r(0)=n,对\[F = (f,\{ {a_i}\} ) \in {\omega _n}\]，定义 \[\begin{gathered} FA{\text{ = }}\int_{{\text{ - }}\infty }^\infty {|f(t) - \sum\limits_{i = 0}^{2n} {{a_i}} } {t^i}{|^2}d{E_t} + \sum\limits_{i = 0}^{2n} {{a_i}} {A^i} \hfill \ DF(A)) = D({A^{2n}}) \cap \{ x \in {\Pi _k}\int_{{\text{ - }}\infty }^\infty {|f(t) - \sum\limits_{i = 0}^{2n} {{a_i}} } {t^i}{|^2}d{\left\| {{E_t}x} \right\|^2} < \infty \hfill \\ \end{gathered} \] 如果f解析，\[F = (f,\{ \frac{{{f^{(i)}}(0)}}{{i!}}\} )\]，那么可得F（A）=f（A）。 定理8 设A是有界自共轭算子，C（A）={0}，r（0）=n，\[G \in {\omega _n}\]，那么 \[\begin{gathered} \overline F (A) = {[F(A)]^ + },(\alpha F + \beta G)(A) = \alpha F(A) + \beta G(A) \hfill \ (FG)(A) = F(A)G(A). \hfill \\ \end{gathered} \] 定理9 设A是\[{\Pi _k}\]上的自共轭算子，C（A）={0}，r（0）=n，\[{F_1} = ({f_1},\{ {a_i}\} ) \in {\Omega _n}\]，\[{F_2} = ({f_2},\{ {a_i}\} ) \in {\omega _n},{f_1},{f_2}\]在\[( - \infty ,\infty )\]连续，在\[\sigma (A)\]上恒等，那么\[{F_1}(A) = {F_2}(A)\]。 定理10 设A是\[{\Pi _k}\]上自共轭算子C（A）={0}，r（0）=n，\[F = (f,\{ {a_i}\} ) \in {\Omega _n}\]f是连续函数，那么\[\sigma (F(A)) = \{ f(t)|t \in \sigma (A)\} \]。 在定理11中，我们建立了F（A）的三角模型并由此证明当\[F = \overline F \]时，\[C(F(A)) = \{ f(t)|t \in C(A)\} \] 定理12 设A施可析\[{\Pi _k}\]空间上的自共轭算子，C（A）={0}，r（0）=n，与0相应的根子空间非退化，T是稠定闭算子，那么\[T \in {\{ A\} ^{'}}\]的充要条件是存在\[F \in {\Omega _n}\]，使T=F（A）。这里\[{\{ A\} ^{'}} = \{ T|\]对满足\[BA \subset AB\]的有界算子B，均有\[BT \subset TB\]}     

Initial Boundary Value Problem for One Class of System of Multi- Dimensional Inhomogeneous GBBM Equations

This paper studies the following initial-boundary value problem for the system of multidimensional inhomogeneous GBBM equations $[\begin{array}{l} {u_r} - \Delta {u_i} + \sum\limits_{i = 1}^n {\frac{\partial }{{\partial {x_i}}}} grad\varphi (u) = f(u),{\rm{ (1}}{\rm{.1)}}\u{|_{t = 0}} = {u_0}(x),x \in \Omega ,{\rm{ (1}}{\rm{.2)}}\u{|_{\partial \Omega }} = 0,t \ge 0,{\rm{ (1}}{\rm{.3)}} \end{array}\]$ The existence and uniqueness of the global solution for the problem(l.l) (1.2) (1.3) are proved. The asymptotic behavior and “blow up” phenomenon of the solution for the problem (1.1) (1.2) (1.3) are investigated under certain conditions.     

我为高考设计题目

题83已知数列{an}为等差数列,数列{bn}为等比数列.(1)若a1+a2+a3=-12,b1·b2·b3=27,且a1+b1,a2+b2,a3+b3是各项均为正整数的等比数列的前3项,求数列{an},{bn}的通项;     

放缩的方法探究

<正>考题(2014年新课标全国卷Ⅱ第17题)已知数列{an}满足a1=1,an+1=3an+1.(1)证明:{an+1/2}是等比数列,并求{an}的通项公式;(2)证明:1/a1+1/a2+...+1/an<3/2.不难证得(1)数列{an+1/2}是以3/2为首项,     

（广义）四元数群的自同构群及其全形

该文讨论了一类2^n阶非交换群——（广义）四元数群Q_{2^n}=〈a,b|a^{2^n-1}=1,b^2=a^{2^{n-2}},{b^{-1}ab=a^{-1}〉(n≥3)的自同构群A（Q_{2^n}）与全形H（Q_{2^n}））的置换表示，给出了A（Q_{2^n}））与H（Q_{2^n}））的构造.     

中立型差分方程解的半环项数的估计

考虑中立型差分方程Δ(xn+pnxn- l) +qnxn- k=0 ,　 n=0 ,1 ,2 ,… ,其中 { pn} ,{ qn}为非负实数列 ,qn>0 .k,l为非负整数 k>l,本文给出其任一解的正、负半环的项数的若干上界估计 .     

我为高考设计题目

题342在数列{a_n}中,若对任意的n∈N^*,都有a_n≤M(实常数)成立,且对任意的aa,则称数列{a_n}具有性质P(M).(1)设等比数列{b_n}(n∈N^*)的前n项和为Tn,若b_3²+b_4=0,b²-2b_3=0;证明:数列{T_n}具有性质P(2);(2)数列{a_n}的前n项和S_n满足:nS_m+n-(m+n)S_n+3(m+n)mn=0(m,n∈N^*);若数列{S_n}具有性质P(884),求a_1的取值集合.     

数学诡辨

(一)X=X+1河南交通学校李丽琴题目:求证劣吕一(2工+1)义=(x十])“一(x+l)(2工+l)证明:将原式两边同晰加上则只须证 2义十1各毛——十1么、下万-~)(x一ZX+l):二〔(、+、卜全全资2〕,两边开平方, 仑x+1 午一即得=(工十l)一 解1:出复数不等式i!:,}一!:川‘1:,全::1簇】:,{a}::!Z}:1、二。可得 !:一。!十!:一3!多l拭一助一;一(:一3)!)11忍川一61二5 故所求匡最小值为5. 解2:由复数不等式 }:‘士::}簇!::{十}::} 可得}:一2卜卜一3}=}:一到十!3一:)}(:一2)+(3一:)}“1 故求的最小位为1. 这岂不是5二1?谁对谁错?万二X十1上期数学诡辩题揭底(二,)…     

CONTINUOUS MAPPING FROM SPHERES TO EUCLIDEAN SPACES

In the forties Knaster, B., posed the following problem: Gieven a continuous mapping f of an (m+n-2) sphere \({S^{m + n - 2}}\) into the Euclidean m -space \({R^m}\） and n distinct points it \({u_1}, \cdots {u_n}\） of \({S^{m + n - 2}}\); does there exist a rotation r such that \[f(r{u_1}) = \cdots = f(r{u_n})?\] In this paper, the index under periodic transfromation of StieM manifold is applied to prove the following theorem: Given a continuous mapping \（f:{S^{k - 1}} \to {R^m}\）， n distinct points \({u_1}, \cdots {u_n} \in {S^{k - 1}}\） viewed as unit vectors satisfying \({u_i}{u_j} = {u_{i + 1}}{u_{j + 1}},i,j \in {I_n}\), and suppose\({u_1}, \cdots {u_n}\） have rank l, then in each of the following cases, there is a!rotation r such that \[f(r{u_1}) = \cdots = f(r{u_n})\] 1. \[n \ne 2,3,k - 1 = (n - 1)m\]; 2. n is an odd prime number, l even,\[k - 1 = \left[ {\frac{{(n - 1)m}}{2}} \right] + l - 2\]; 3. n is an odd prime number, l odd, \[l \ge \left[ {\frac{{(n - 1)m}}{2}} \right] + 1,k - 1 = \left[ {\frac{{(n - 1)m}}{2}} \right] + l - 2;\] 4. n is an odd prime number, l odd, \[l < \left[ {\frac{{(n - 1)m}}{2}} \right] + 1,k - 1 = (n - 1)m + 1;\] where [*] is the least even number>*. This theorem generalizes the classical Borsuk-Ulam theorem.