SOME FIXED POINT THEOREMS OF CONTRACTIVE TYPE MAPFINGS

1. Let X be the conjugate of a separable Banach space satifying the *-Opial condition, i. e., if \[\{ {x_n}\} \subset x,{x_n}\mathop \to \limits^{{w^*}} {x_\infty },{x_\infty } \ne y\], then\[\mathop {\overline {\lim } }\limits_{n \to \infty } ||{x_n} - {x_\infty }|| < \mathop {\overline {\lim } }\limits_{n \to \infty } ||{x_n} - y||\] for rxample \[X = {l_1}\] Let K be a nonempty weak* closed convex subset of X. The main results are: Theorem 1. Suppose T is a ooniinuons mappings of K into itself such that for every \[x,y \in K\]，\[||Tx - Ty|| \le a||x - y|| + b\{ ||x - Tx|| + ||y - Ty||\} + c\{ ||x - Ty|| + ||y - Tx||\} \] where real numbers \[a,b,c \ge 0\] and \[a + 2b + 2c = 1\]. Suppose also K is bounded.Then T has at least one fixed point in K. Theorem 2. Let T be a mapping of K into itself, and \[a(x,y),b(x,y),c(x,y)\]be real functions such that for all\[x,y \in K\] \[||Tx - Ty|| \le a(x,y)||x - y|| + b(x,y)\{ ||x - Tx|| + ||y - Ty||\} + c(x,y)\{ ||x - Ty|| + ||y - Tx||\} \] and \[a(x{\rm{y}},y){\rm{ + }}2b(x,y){\rm{ + }}2c(x,y) \le 1\] Suppose there exists \[x \in K\] such that \[O(x) = \{ {T^n}x\} _{n = 1}^\infty \] is bounded and \[\mathop {\inf }\limits_{y,z \in o(x)} c(y,z) > 0\] Then T has at least one fixed point z in K and \[{T^n}x\mathop \to \limits^{{w^*}} z\]. 2. We denote \[CL(x) = \{ A;nonempty{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} closed{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} subset{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} of{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} X\} \] \[K(x) = A;nonempty{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} closed{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} subset{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} of{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x\} \] here X is a complete metric space with metric d. On \[CL(x)\] and \[K(x)\] we introduce the generalized Hausdorff distance \[H(,)\], The main results are: Theorem 3. Suppose \[\{ T,S\} \] is a pair of set-valued mappings of X into \[CL(x)\],which satisfies the following condition: \[H(Tx,Sy) \le hMax\{ d(x,y),D(x,Tx),D(y,Sy),\frac{1}{2}[D(x,Sy) + D(y,Tx)]\} \] for each \[x,y \in K\], where 0相似文献   

Quasi-Normal Structure and Fixed Point of Nonlinear Mappings

Let (X,|| ||) be a Banach space. For $\Omega \subset X^*$ and $x\in X$ we introduce the following notations (p\geq 1 and n\in N) $|X|_{\Omega _p(n)}=sup{(\sum\limits_{f\in F} |f(x)|^p)^{1/p}:F \subset \Omega,|F|\leq n$ $|X|_{\Omega _\infty}=sup{|f(x)|:f\in \Omega}$ A convex subset E of X is said to have guasi-normal structure whenever there exists a norm 1 | on A which satisfies the following conditions; (i) E has norinal structure relative to the norm ||| |||. (ii) There exist $\Omega \subset X^*$, p\geq 1 and \theta \in (0,1], such that $|x|_{\Omega _p(2) \leq |||x||| \leq ||x||}$ for x\in E and |||x|||<||x|| implies $2^1/p |x|_\Omega_\infty \geq \theta ||x||+(1-\theta)|||x|||$ or (ii)' There exist \Omega \subset X^*,p\geq 1 and \alpha \in [1,4^1/p) such that for all x\in E, |x|_\Omega_\rho(4)\leq |||x|||,||x||=max{|||x|||,\alpha|x|_\Omega_\infty} and for any countable subset w of \Omega $sup{\sum\limits _{\delt\in w |f(x)|^p:x\in E}<+\infty$ We notice that a set with normal steucture must have quasi-normal structure and there exist sets without normal structure which quasi-normal structure. The main result of the present paper is as follows. Theorem. Let (X, || ||) be a Banach space, E a weak compact convex nonempty subset of X with quasi-normal structure. Let T be a mapping of E in to itself. If there exists a sequence {x_n} in any T-invariant convex subset of E such that $lim_{n\rightarrow \infty} ||x_n-x_n+1||=lim_{n\rightarrow \infty}||x_n-Tx_n||=0$ and $lim_{n\rightarrow \infty} ||y-x_n||=\delta(\bar co{x_n}),for y\in \bar co({c_n})$ limll2/-*?ll=3(coK}), for y€co({xa}), then the mapping T has a fixed point in E, In particular, if the mapping T satisfies $||Tx-Ty||\leq max{||x-y||,1/2(||x-Ty||+||y-Tx||)},for x,y\in E$ then the mapping T has a fixed point in E.     

机会约束规划中若干凸性命题

在随机规划中,机会约束规划的一般形式是: m;n华(二). 5 .t .P(。!A(。)、》b(。)))a,o镇a成l,二〔X.或者m in5 .t.甲(二),P(。{月‘(。)x)b‘(。)))a‘,o戒a‘(I,二任X.其中甲(劝是凸函数,X是户‘上的凸集,月(。)是。xn维矩阵,月‘(。)是A(。)的第‘个行向量,6(。)是。维r句量,b‘(。)是b(。)的第i个分量,且(。),A‘(。),b(。),b‘(。)的元素是定义在概率空间(口,J,尸)上服从已知(联合)分布的随机变量或常量.机会约束规划时主要问题之一是可行集 X(a)二{x IP(。}A(。)x》b(。))》a,x〔X}或 X‘(a‘)={二}P(。}月‘(。)x>b‘(。))>a,…     

2、3能被2整除

3能被2整除,这显然是错误的,但是有人却能证明出来。不信,请石证明:rl一等比数列{x“一‘}rfJ前,:项和公式1 x xZ … x一1=l二Lx: 1一X(x笋1,。〔N)有I一x“=(1一x)(1十x十护 …x’一, a.(1一x’)=。’一王(a一。x)(1 x *2 …x“一‘)令ax=b(。法,.。子b),贝!}a一b‘=a‘一’(a     

关于三次样条函数界值问题的注记

对区间〔a,如上的分划△:a二x。…     

A DIRECT SEARCH METHOD BY THE LOCAL POSITIVE BASIS FOR LINEARLY CONSTRAINED OPTIMIZATION

讨论下列线性约束最优化问题 \[LNP{\kern 1pt} {\kern 1pt} {\kern 1pt} \mathop {\min }\limits_{{\text{x}} \in X} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{f}}(x),X = \{ x|x \in {R^n},{({a^i})^T}x \geqslant {a_i},i \in {I_m}\} \] 其中\[{I_m} = \{ 1,2,...,m\} \]，对于X中的能行点，定义了局部能行锥与相应的局部正基一 即生成该锥的一组正独立的向量，给出了沿着局部正基方向进行目标函数值比较与迭代 点移动的算法模型，简称为局部正交基方向搜索法，本文并证明了这算法的收敛性定理： 定理 设约束集合\[X = \{ x|{({a^i})^T}x \geqslant {a_i},i \in {I_m}\} \]非空有界且非退化，目标函数f(x)连续可微，{yi}是局部正基方向搜索法产生的某个点列，那么{yi}的任意极限点x*必是问题（TNP）的Kuhn-Tucker点。     

ON PGLAR PRODUCT OPERATOR

Let H be a Hilbert space, and let A be a linear bounded operator on H. For \(\lambda \in \rho (A)\), the \({U_\lambda } = {(A - \lambda )^{ * - 1}}(A - \lambda )\) is called polar.Produot operator. In this paper, we discuss the properties of \({U_\lambda }\) and the relation between \({U_\lambda }\) and A. We obtain tbe following results. Definition. Let B be a linear bounded operator on H, suppose \(0 \in \rho (B)\). For every \(x,y \in H\), we definite \([x,y] = (Bx,y)(H,B)\)(or (H, [·,·]) is called a non- degenerate bilinear space (it is obvious that if B=B*,then (H,B)is a space with an indefinite metric; and that if B>0, then (H,B) is a Hilbert Space. If an operator U(A) satisfies \[[Ux,Uy] = [x,y]([Ax,y] = [x,Ay]),x,y \in H\] then the operator U(A) is called a wvitary (self adjoint) on (H,B). Theorem I . Suppose A is a linear bounded operator on H, (1) If \(0 \in \rho (A)\), then \(U = {A^{ * - 1}}A\) is a unitary operator on (H,A) or (H, A*), and \(\sigma (U) = \frac{1}{{\sigma (U)}}\). (2) If there is a complex number \(\alpha \), such that \({\mathop{\rm Im}\nolimits} A \ge \alpha > 0\) then a)\(0 \in \rho (A)\), and the operator \(U = {A^{ * - 1}}A\) is a unitary on Hilbert space \((H,{\mathop{\rm Im}\nolimits} A) and 1 \in \rho (U)\);b) there exist two Hilbert spaces \((H,{v_1}),(H,{v_2})\), such that A, A* are all the unitary operator from (H,v1) onto (H,v2), and there are two spectral measures \(\{ E_\lambda ^i,\lambda \in [\alpha ,\chi ] \subset (0,2\pi )\} ,i = 1,2,\), such that \(AE_\Delta ^1H \subset E_\Delta ^2H,{A^ * }E_\Delta ^1H \subset E_\Delta ^2H\) for any \(\Delta = (\lambda ,u] \subset (0,2\pi ]\). (3) If \(0 \in \rho (A) \cap \rho ({\mathop{\rm Im}\nolimits} A)\) then the operator \(U = {A^{ * - 1}}A\) is a unitary on \((H,{\mathop{\rm Im}\nolimits} A)\). with an indefinite met He, and \(1 \in \rho (U)\). (4) For any complex number \(\lambda = r{e^{i\theta }},\left| \lambda \right| > \left\| A \right\|\), then \({U_\lambda }\) must be a unitary operator on the Hilbert space \(\left( {H,{\mathop{\rm Im}\nolimits} (\frac{1}{{i\lambda }}A + iI)} \right),and - {e^{i2\theta }} \in \rho ({U_\lambda })\) Theorem 2. (1) A is a normal operator iff there exists a complex number \(\lambda \), \(\lambda \in \rho (A)\), such that \(\frac{{\partial {U_\lambda }}}{{\partial \lambda }}{U_\lambda } = {U_\lambda }\frac{{\partial {U_\lambda }}}{{\partial \lambda }}\),where \(\frac{\partial }{{\partial \lambda }}\) is the directional derivative. (2) If there exists a complex number \(\alpha ,{\mathop{\rm Im}\nolimits} A \ge \alpha > 0\) then A is a normal operator iff \(U = {A^{ * - 1}}A\) is also. (3) If A is a hyperiwrmal or a subnormal, then for every \(\lambda \in \rho (A),\sigma ({U_\lambda })\) lies on the circle. 3, 4期 关于极?积算子炉一U 499 Theorem 3? Let A be a linear bounded operator on Hm Suppose 0￡p(J.)? (1) If U=А*~гА is a unitary operator in a certain non-degenerate bilinear space (H} 1 J |Л|>1 入 J Ш=1 solmble iff (l)cr(ü) =-=i=-, (2) the operotor Ui= f XdE\ is unitarity equivalent to cr(C7) J |л|>1 the operator ül_1 == ШЕ1, 、 J 1Л|>1 If the conditions (1),(2) are satified, then we have (3) the subspace JJi?^2 of H reduces any solution of the equation A*"1 A — U, and 1 AI (Ягея*)1 = ^Uo where Ъ is any in vertible self-adjoint on (i?i?J?2) L, and bv Uo (Uo = Jiai i ЫЕ1 )}⑷41Я1фя, = , Ла = A1ü2, if the operator V is to realize TJt and Ut^unitary equivalence^ then Ai = VSf where S is any invertible on Нг and SvUi. " Corollary. Swpp)se operator U is a normal on a certain Hilbert space(Hy v) (^the U is similar to a certain normal operator on IT), if U satisfies the conditions (1)、(2) of Theorem^ on ?S,v) ? Then the general form of the solution of А*~гА = U is A = vA'y where A' is same as A in Theorem^. Theorem 5. Let U be a linear bounded operator on H. Suppose O?p(J.)and p(ü) is a simply connected region, then the equation А*~гЛ = U is a solvable iff there exists a certain space {H,v) with a indefinite metric, such that U is a unitary operator on (Hf 丨）. If is a unitary operator on ?H,v),then there exists a particular solution of I X+$ А*~гА = U: Ar = 2e v [ (?7 —X)_1+X], where eie ? p (JJ), and the general form of the solution is Аж АУ, wliere V is any in vertible self-adjoint on (H, -u)and VvU,     

关于函数定义的选择题

1.下列方程式中,同时又是函数筋析式的是 (A)岁==士x(刀)y=t xl (e)1 01二x(D),=“,一}二}一: 2.下列每对函数中,相同的函岑对是 (A)万二一2:与梦=忍}盆( (B)夕二2}戈,戈夕二乞一:l (C)刀二2’与,二:}匕’{ (D)万=:又l与犷一}乙’I 3.下歹叮函器甲.存在反函赘、的是夕=护,x〔〔一1,1〕2!,}x〔Ry=}l(g挤夕二歇n丫,x任,汀任(0 .1)(O,劝。(A)(B)(C)(D)附:1。上期本栏答案【:D;召l?.。︸6DC33关于函数定义的选择题@杨晓红$黑龙江海伦教师进修学校~~…     

QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS ON THE PROJECTIVE PLANE

Consider the differential system \[\frac{{dx}}{{dt}} = P(x,y),\frac{{dy}}{{dt}} = Q(x,y)\](1) where P(x, y), Q(x,y) are defined on the square S: [0, a]X[0, a], continuous and have continuous first partial derivative there, and satisfy the following relations P(0, y)=P(a,a-y), Q(0, y) = -Q(a, a-y), P(x, 0) = —P(a-x,a), Q(x,0)=Q(a-x, a), \[(x,y) \in [0,a] \times [0,a]\], A projective plane will be viewed as the square S in the (x,y) -plane, in which the points (0, y), (a, a -y) or (x, 0), (a—x, a) on opposite sides of the square are identified. Thus, under condition (2), (1) is a differential system defined on the pro?tective plane. On the projective plane, in addition to closed curves in the usual sense (we call it 0-closed curve), there are also closed curves consisting of several arcs in the square, such closed curves are illustrated in Fig. 1 (in which we use arrows and numbers to show that a closed curve can be constructed according to this direction and order). Hereafter, we will call the closed curve JT on the projective plane an closed curve, if .T consists of n arcs which do not meet each other in S, and each arc intersects the sides of 8 at its two end points only. If n is even (odd), then we also call P an even- closed curve (odd-closed curve) on the projective plane. Lemma 1. An even-closed curve on the projective plane divides the projective plane into two parts, but an odd-closed curve does not. So we can define in a certain sense the interior and exterior of an even-closed curve, while for an odd-closed curve, we can not define its interior and exterior. We call L left-right orientead family of directed arcs in S, if the origin of every arc in L is at the left hand side of its end. Similarly, we can define right-left, upper- lower and lower-upper oriented families.? Lemma 2. Let \[\Gamma \] be a closed orbit of system (1) on the projective plane consisting of only oriented ares of the same kind, then \[\Gamma \] contains two arcs atmost. On the projective plane, we can define limit cycle of the differential system (1) as in [1]. In particular, we can define stable and unstable cycles as well as semi-stable cycle for an even-closed orbit, but if an odd-closed orbit is a limit cycle, it must be a stable or unstable limit cycle. Let us extend system (1) to the square S*: [—a, a] x [—a, a] by defining first in [-a, a] X [0, a]: \[\begin{gathered} {P_1}(x,y) = \left\{ {\begin{array}{*{20}{c}} {p(x,y),(x,y) \in [0,a] \times [0,a]} \\ {p(a + x,a - y),(x,y) \in [ - a,0] \times [0,a];} \end{array}} \right. \hfill \ {Q_1}(x,y) = \left\{ {\begin{array}{*{20}{c}} {Q(x,y),(x,y) \in [0,a] \times [0,a]} \\ { - Q(a + x,a - y),(x,y) \in [ - a,0] \times [0,a];} \end{array}} \right. \hfill \ \hfill \\ \end{gathered} \] and then in s*: \[\begin{gathered} {P_2}(x,y) = \left\{ {\begin{array}{*{20}{c}} {{p_1}(x,y),(x,y) \in [ - a,a] \times [0,a]} \\ { - {p_1}( - x, - y),(x,y) \in [ - a,a] \times [ - a,0];} \end{array}} \right. \hfill \ {Q_1}(x,y) = \left\{ {\begin{array}{*{20}{c}} {{Q_1}(x,y),(x,y) \in [ - a,a] \times [0,a]} \\ { - {Q_2}( - x, - y),(x,y) \in [ - a,a] \times [ - a,0];} \end{array}} \right. \hfill \\ \end{gathered} \] It is easily seen that \[\frac{{dx}}{{dt}} = {P_2}(x,y),\frac{{dy}}{{dt}} = {Q_2}(x,y)\](6) is a C1 differential system on the torus formed by identifying opposite sides of S*. A closed orbit of (1) on the projective plane must correspond to some closed orbits of (6) onthe torus. We can prove now the following theorems. Theorem 1. Let \[m = {m_1} \cup {m_2}\] and \[n = {n_1} \cup {n_2}\] be two 2-closed curves in the projective plane, and n is in the interior of m. Suppose the domain Q bounded by m and n contains no stationary points., and trajectories of (1) crossing m all run from exterior to interior, while trajectories crossing n all run from interior to exterior. Then Q contains at least two 2-closed orbits F and L, where F is outer-stable, L is inner- stable limit cycle. Here r may coincide with L, if this takes place, then F=L is a stable limit cycle. Theorem 1.1. Let m be a 2-closed curve in the projective plane which consists of arcs joining opposite sides of the square 8. The interior of m contains no stationary points, and trajectories crossing m all run into the interior of m, then in the interior of m there is at least a closed orbit of (1) which is an outer stable 2-limit cycle or a stabe 1-limit cycle. Theorem 1.2. Let \[\Gamma \] be a 2-closed orbit of (1) in the projective plane which consists of arcs joining opposite sides of the square S. The interior of \[\Gamma \] contains no stationary points, then in the interior of \[\Gamma \] there is a 1-closed orbit. Theorem 3. Let Q be a domain in the projective plane, and B(x, y) be a single- valued continuous function in G which has continuous first partial derivatives. Suppose \[\frac{\partial }{{\partial x}}(BP) + \frac{\partial }{{\partial y}}(BQ)\] does not change its sign in G, and the set \[\frac{\partial }{{\partial x}}(BP) + \frac{\partial }{{\partial y}}(BQ) = 0\]contains no 2-dimensionaL domain, then the system (1) has no even-closed orbit whose interior is in G. In particular, if G is the whole projective plane, then the system (1) has no closed orbit at all. Theorem 4. Suppose F(x, y) = C is a family of curves, where F(x, y) is a single valued continuous function and has continuous first partial derivatives in the projective plane. \[P\frac{{\partial F}}{{\partial x}} + Q\frac{{\partial F}}{{\partial y}} does not change its sign in a domain G, and the subset of G in which \[P\frac{{\partial F}}{{\partial x}} + Q\frac{{\partial F}}{{\partial y}} = 0\] contains no closed orbits of (1), then system (1) II. Examples The systems in the following examples are all systems of differential equations (defined in the projective plane, and the projective plane is formed by the square [0, pi] x [0, pi]. A, B, C, D are constants other than zero. \[Ex.1{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{dx}}{{dt}} = A\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2x + B\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} y,\frac{{dy}}{{dt}} = C\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 4y\] Using theorem 4 and theorem 1, we can prove that (8) has a 2-olosed orbit and an 1-closed orbit, which are stable and unstable limit cycles respectively. \[Ex.2{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{dx}}{{dt}} = B\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} y,\frac{{dy}}{{dt}} = C\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x + \frac{D}{2}\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2y\] system (9) has always a 1-closed orbit, if \[\left| {\frac{D}{{2C}}} \right| > 1\], then we can prove that (9) has no 2-closed orbit. \[Ex.3{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{dx}}{{dt}} = \frac{A}{2}\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2x,\frac{{dy}}{{dt}} = C\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x + \frac{D}{2}\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2y\] where D>-2C>0 This system has ho closed orbit, although the projective plane is a one-sided surface.     

RETRACTED ARTICLE: Some applications of BP-theorem in approximation theory

In this paper we apply Bishop-Phelps property to show that if X is a Banach space and G X is the maximal subspace so that G⊥ = {x* ∈ X*|x*(y) = 0; y∈ G} is an L-summand in X*, then L1(Ω,G) is contained in a maximal proximinal subspace of L1(Ω,X).     

关于L~1收敛的若干定理

引言设S。(x)=习akeo。欠:,D。(x)=生+2艺eos欠x,“(.’’(x)=击燕“兮’‘x),其中 斗氏一2s梦,(x)==艺七’ak 西.备0,e。,(、二十华),当s。(x)收敛时;己其极限为,(x),假如存在。>。,使 、乙/”一口”、les.…l甘.lse、.flwelesesl.esl沙BV一,)那末称数列(q户是拟单调的。对r=卜,:客“·,么一,相似文献   

THE ASYMPTOTIC PROPERTIES OF WEIGHTED MARKOV OPERATORS

IInttodllct1OllLet(X,E,u)be a a－inite me。ure svace．L‘＝{f:人*(x)I。(d。)＜co}．D＝{j。L‘:f＞0川f【＝1};any function f E D is called a density,and。j(A)＝jA f(x)。(dx)(A E Z)isthe Prob——hi,measurt。Sociattd WZth/．/。*‘,…0＝人/(t)…dt);j“()＝ti(厂 j(t》;and f－(x)＝max(0,一/(x))．By the support of a function g E L‘we simply mean the set ofall。such that g(x)一 0,that is,suppo＝《x:g(x)f 0}．Notice that suppg Is unique up to azero measure s戌．A s虹 M C LI Is said to be wea…     

上期课外练习参考解答

!初一年级}1.’:艺3一/5一乙7一45“, 艺1 艺9一/2 艺6一乙4 乙8一90”, 匕l 匕2 匕3 艺4 艺5 乙6 艺7 艺8 乙9 =3X45。 3X900=4050.2.过A作AD// BC交圆A于D,延长BA交圆A于 E,则艺DAC一艺C,艺EAD一匕B.公共部分的面 积之和为一个半圆的面积. 设圆的半径为二cm,则喜二xZ一8二,,2一16, 乙②③ 工一4.故圆的半径为4cm. “ b一c d, (a b)2=(c d)2.即aZ Zab bZ=。2 dZ Zed.又aZ bZ=cZ 己,, ab一cd. (a一b)2=aZ bZ一Zab 一c2 dZ一Zcd一(c一d)2. a一b一c一d或a一b一d一c①若③成立,则有a=。,b*d.着③成立,则有a=d,b二‘..,.总有aZo…     

线性规划的新解法

号1准备工作在展开线性规划问题的新解法的讨论之前,先将要用到的工具,提前说明或证明出来。引理1在E·中,任意。十1个矢量日,武…尿 .有下述代数恒等式成立:艺(一l)‘〔a.,一,今~咔及2…在k_‘a*,a* .,…a。十t〕a*“0换言之其中:当〔a、,aZ…a。〕手。时,有a。,.“艺j,a,占*=(一l)二〔鼠,矶,a卜‘,a‘-武、:…砚 :〕/〔拭,矶,…斌(证明略去)在E”中,矢量斌,·澎构成一个n阶矩阵,记为:口一,口之一alZ,…a.。CZ之,一aZ,(a,,aZ,a,l,a.2一‘”a。,这里a‘- 当斌 口,。)a。是尸卜独立矢量时有且必有det(鼠,矶,…,武)羊。. 竹. ‘..a。 , 自…     

REVERSIBLE MARKOV PROCESSES IN ABSTRACT SPACE

Let \[(E,{\cal E})\] be a measurable space and every single point set {x} belong to \[(E,{\cal E})\].\[q(x) - q(x,A)(x \in E,A \in {\cal E})\]is said to be a q-pair, if (i) For fixed i，q(°), Л)\[A,q(),q(,A)\] is a \[{\cal E}\]-measurable function of x; (ii) For fixed \[x,q(x, \cdot )\] is a measure on \[{\cal E}\], and \[\begin{array}{l} 0 \le q(x,A) \le q(x,E) \le q(x) < \infty .(\forall x \in E,\forall A \in {\cal E})\q(x,\{ x\} ) = 0,(\forall x \in E) \end{array}\] A q-pair of furiotions q (x)- q (x,A) is called conservative when \[q(x,E) = q(x),(\forall x \in E)\]. \[{P_t}(x,A)(t \ge 0,x \in E,A \in {\cal E})\] is said to be a q-process, if (i) For fixed t, A, \[{P_t}(x,A)\] is a \[{\cal E}\]-measnrable function of x; (ii)For fixed t,x, \[{P_t}(x, \cdot )\] is a measure on ê\[{\cal E}\] and \[0 \le {P_t}(x,E) \le 1\]; (iii) \[{P_{s + t}}(x,A) = \int_E {{P_t}} (x,dy){P_s}(y,A),{\kern 1pt} {\kern 1pt} {\kern 1pt} (x \in E,A \in {\cal E},t,s \ge 0)\] (iv)\[\mathop {\lim }\limits_{t \to {0^ + }} \frac{{{P_t}(x,A) - {I_A}(x)}}{t} = q(x,A) - q(x){I_A}(x)(\forall x \in E,\forall A \in {\cal E})\] It is called honest when \[{P_t}(x,E) = 1,{\kern 1pt} {\kern 1pt} {\kern 1pt} (\forall t \ge 0,\forall x \in E)\] A q-process \[{{P_t}(x,A)}\] is called reversible, if there is a probability measure \[\mu \] on \[{\cal E}\] such, that \[\int_A {\mu (dx){P_t}} (x,B) = \int_B {\mu (dx){P_t}} (x,A){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (\forall t \ge 0.\forall A,B \in {\cal E})\] In this paper, we obtain some oriterions for (i)The existence of a reversible q-process;. (ii)The existence of a honest reversible q-process； (iii)The uniqueness of reversible q-process when the q-pair is conservative.     

METRIC ENTROPY OF HOMEOMORPHISM ON NON-COMPACT METRIC SPACE

Let T : X → X be a uniformly continuous homeomorphism on a non-compact metric space (X, d). Denote by X* = X ∪ {x*} the one point compactification of X and T * : X* → X* the homeomorphism on X* satisfying T *|X = T and T *x* = x*. We show that their topological entropies satisfy hd(T, X) ≥ h(T *, X*) if X is locally compact. We also give a note on Katok’s measure theoretic entropy on a compact metric space.     

在Banach空间中推广的 Roper-Suffridge算子(III)

假定 $X$ 是具有范数$\|\cdot\|$的复 Banach 空间, $n$ 是一个满足 $\dim X\geq n\geq2$的正整数. 本文考虑由下式定义的推广的Roper-Suffridge算子 $\Phi_{n,\beta_2, \gamma_2, \ldots , \beta_{n+1}, \gamma_{n+1}}(f)$: \begin{equation} \begin{array}{lll} \Phi _{n, \beta_2, \gamma_2, \ldots, \beta_{n+1},\gamma_{n+1}}(f)(x) &;\hspace{-3mm}=&;\hspace{-3mm}\dl\he{j=1}{n}\bigg(\frac{f(x^*_1(x))}{x^*_1(x)})\bigg)^{\beta_j}(f''(x^*_1(x))^{\gamma_j}x^*_j(x) x_j\\ &;&;+\bigg(\dl\frac{f(x^*_1(x))}{x^*_1(x)}\bigg)^{\beta_{n+1}}(f''(x^*_1(x)))^{\gamma_{n+1}}\bigg(x-\dl\he{j=1}{n}x^*_j(x) x_j\bigg),\nonumber \end{array} \end{equation} 其中 $x\in\Omega_{p_1, p_2, \ldots, p_{n+1}}$, $\beta_1=1, \gamma_1=0$ 和 \begin{equation} \begin{array}{lll} \Omega_{p_1, p_2, \ldots, p_{n+1}}=\bigg\{x\in X: \dl\he{j=1}{n}| x^*_j(x)|^{p_j}+\bigg\|x-\dl\he{j=1}{n}x^*_j(x)x_j\bigg\|^{p_{n+1}}<1\bigg\},\nonumber \end{array} \end{equation} 这里 $p_j>1 \,( j=1, 2,\ldots, n+1$), 线性无关族 $\{x_1, x_2, \ldots, x_n \}\subset X $ 与 $\{x^*_1, x^*_2, \ldots, x^*_n \}\subset X^* $ 满足 $x^*_j(x_j)=\|x_j\|=1 (j=1, 2, \ldots, n)$ 和 $x^*_j(x_k)=0 \, (j\neq k)$, 我们选取幂函数的单值分支满足 $(\frac{f(\xi)}{\xi})^{\beta_j}|_{\xi=0}= 1$ 和 $(f''(\xi))^{\gamma_j}|_{\xi=0}=1, \, j=2, \ldots , n+1$. 本文将证明: 对某些合适的常数$\beta_j, \gamma_j$, 算子$\Phi_{n,\beta_2, \gamma_2, \ldots, \beta_{n+1}, \gamma_{n+1}}(f)$ 在$\Omega_{p_1, p_2, \ldots , p_{n+1}}$上保持$\alpha$阶的殆$\beta$型螺形映照和 $\alpha$阶的$\beta$型螺形映照.     

吉林大学一九八四年硕士研究生入学试题

J目.J.J不‘孟‘‘斌.,,邵汤气r日了梦孙、叮弓七』:卜.J今JJ‘2〔一’考虑边值问题 g:,,,口“子_‘。八口“u、口f,,.八口u、._,__、__,,__、 龟去‘二贡t乞气叭万)介方一j一兰一lb〔x)芒井}+c‘二):=厂(x),0蕊x(l, 1口x‘\口x‘,口x\口x/ 才‘,_日U_。、,,,_八1 了“二卫二一“O。当x=0 .1。 又口x‘一‘这里a(x)任C“(〔0,l」),西(x)任C‘(〔0,l]),C(x扩(x)任C“(巨0,l〕),a(x))a。>0,。。几级一‘数,b(x),c(x))0.试给出并证明和它相应的极小位能原理.(20分)二、试确定求积公式 J{。,‘X)dX澎‘{·,(一,卜。,(。卜·,(、)}中的系数…     

Analytic Operator Rings and Analytic J-Self Adjoint Operator Ring

Let A be a linear bounded operator in Hilbert space H with polar respresentation A =J(A^*A)^{1/2} where J^2=I, J^* = J. we use \pho_J(A) to denote the set of all complex \lambda, such that for any $\lambda \in \rho_J(A)$ there exist an bounded inverce R_J(A,\lambda) of (A—\lambda J)and \sigma_J(A)to complement of \rho_J(A). Let S be a closed Cauchy domain, S\supset \sigma_J(A) and f (z) an analytic function on S. We define $f(A)=\frac{1}{2\pi i}\oint\limits_{2s} {f(\zeta ){R_J}(A,\zeta )d\zeta }$, the set of all such f(A)is denoted M. If f(z) be analytic on S and symmetrical for real axis then f(A) is J-self adjoint. The set of all such f(A)is denoted M'. Let A\otimes B = AJB for A, B \in M(or M'). We have Theorem, the ring of functions analytic on S (or analytic symmetrical for real axis on S) is a algebra homomorphism of M (or M'). The constant function 1 or z corresponds to operator J or A^* respectively. Let $M_J={JB|B \in M}$ and $M'_J={JB|B \in M'}$ If the spectrum of (A^*A)^{1/2} is detached, we have Theorem. M_J has common non-trivial reducing subspace and it is true for M_J.     

2005年高考数学广东卷

一、选择题’ 1.若集合人健‘(x日xl成2},N一{刘尹一3x一0}, 则M门N=() A.{3}B.(0)C.{0,2}D.{0,3} 2.若(a一2泛)i二吞一i,其中a、b任R,￡是虚数单 位,则aZ+b之=() D.5 3 .lim 男‘.一3 x十3 xz一9 ②若m、l是异面直线,l//a,m//a,且n上l, n上m,则n上a; ③若l//a,m//月,a//月,则l//m; ④若l泣a,m仁a,l门m=点A,l//月,m//月,则 a//尽 其中为假命题的是() A.①B.②C.③D.④ 8.先后抛掷两枚均匀的正方体般子(它们六个 面分别标有点数1、2、3、4、5、6),散子朝上的 面的点数分别为X、Y,则logZxy一1的概率 为() 5一,2 A.0 B.ZC A一含B·O…