关于Gauss过程增量的若干结果

Let {X(t);t≥0}be a Gaussian process with stationary increments,X(0)=0(a.s.),EX(t)=0 andσ~2(h)=EX(t+h)-X(t)~=EZ~2(h)=Coh~(2α),0<α≤1.In this paper,we first prove that the Levy's theorem of the modulus of continuityof the Wiener process is also true for {X(t);t≥0};i.e.Furthermore,we point out that some results on increments of the Wiener processesin[3]and[4]remain true for the increments of {X(t);t≥0}.     

Banach空间中渐近非扩张映射的不动点迭代

设D是一致凸Banach空间X的非空闭凸子集 ,T∶D→D是渐近非扩张映射且kn ≥ 1 ,∑ ∞n =1(kn- 1 ) <∞ .设T的不动点集F(T) ≠ ,T是全连续的 (X满足Opial条件 ) ,{xn},{yn},{zn}由定义 2给出 ,如果 ∑∞n =1cn <∞ ,∑ ∞n =1c′n <∞ ,∑ ∞n =1c″n <∞ ,且下列条件之一满足 :(i)b″n ∈ [a ,b] ( 0 ,1 ) ;b′n ∈ [0 ,β];bn ∈[0 ,α],αβ β <1 ;(ii)b′n ∈ [a ,b] ( 0 ,1 ) ;b″n ∈ [a ,1 ];bn ∈ [0 ,b];(iii)bn ∈[a ,b] ( 0 ,1 ) ;b′n ∈ [a ,1 ],则 {xn},{yn},{zn}强收敛于T的不动点 .( {xn}弱收敛于T的不动点 ) .     

A BOUNDARY VALUE PROBLEM FOR A NONLINEAR ORDINARY DIFFERENTIAL EQUATION INVOLVING A SMALL PARAMETER—The Riemann problem for a Generalized Diffusion Equatiois

This paper studies the boundary value problem involving a small parameter $$((k(V(t))+s)|V'(s)|^{N-1}V'(s))'+(sg(V(s))+f(V(s)))V'(s)=0 for s\in R$$, $$V(-\infty)=A,V(+\infty)=B;A0$$, $$U(x,0)=A for x<0,U(x,0)=B for x>0$$ under the hypotheses H1—H4 . The author's aim is not only to determine explicitly the discontinuous solution ,to the reduced problem；and the form and the number of its curves of discontinuity, but also to present, in an extremely natural way, the jump conditions which it must satisfy on each of its curves of diseontinuity. It is proved that the problem has a unique solution $U_{\varepsilon}(x,t)=V_{\varepsilon}(s),s=x/p(t),s\geq0,V_{\varepsilon}$pointwise converges to $V_{0}(s)$ as $s\downarrow0,V_{0}(s)$ has at least one jump point if and only if k(y) possesses at least one interval of degeneracy in [A-B], and there exists a one-to-one correspondence between the collection of all intervals of degeneracy of k(y) in [A-B] and the set of all jump points of $V_{0}(s)$     

准马尔可夫矩阵能扩充为转移矩阵的充要条件

<正> 一个取值于{0,1,2,…,N}的随机过程 Y(t)(t≥0) 称为 n 阶准马尔可夫链,如果对任意 i=1,2,…,N,T>0,在事件{Y(T)=i}和(?)_T={Y(s);0≤s≤T}的条件下,过程 Y(T+t) (t≥0) 的有限维分布仅依赖于 i 而不依赖于 T 和(?)_T(见[1]).当此性质对 i=0也成立,Y(t)就是通常的马尔可夫链.     

On Applications of Vector Measure to the Optimal Control Theory for Distributed Parameter Systems

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 ].     

REGULARITY OF SUPERMARTINCALES INDEXED BY A LATTICE-ORDERED SET

Let X= {x_t; t∈T} be a supermartingale (resp. martingale) defined on a probability space (Ω,(?), P) with respect to (?), where (?)={(?)_t; t∈T} is nondecreasing family of sub-σ-algebras of (?). It is well known that (see Theorem 2.27 in [1] ) when T=(?)= {0, 1, …,∞}, the supermartingale (resp. martingale) X possesses the (?)-regularity, i.e., for every τ,σ∈(?) such that τ≤σ,     

依中间意义渐近非扩张的半群的几乎轨道的渐近行为

设X是具有Frechet可微范数的一致凸Banach空间,C是X的非空有界闭凸子集,T={T(t):t≥0}是C上依中间意义渐近非扩张的半群。若μ(·):[0,∞)→C是T={T(t):t≥0}的几乎轨道且关于t∈[0,∞)连续,则{μ(t):t≥0}几乎弱收敛到集合∩_(t>0)co{μ(r):r≥t}∩F(T)的唯一点。     

用二进制解2003年(全国卷)理科压轴题

题　设 { an}是集合 { 2 t+2 s|0≤ s相似文献   

THE STRONG LAW OF LARGE NUMBER AND PARAMETER ESTIMATION OF ONE CLASS OF NON-NEGATIVE INTEGER-VALUED TIME SERIES

1.IntroductionLet{xt}beatimeseriesdefinedonprobabilityspace{fi,F,P),satisfyingwhere(1)p(t,s,j)isarandomfieldwithValueIor0,t,8EI={0,11,f2,'.},jEN={1,2,'.},andsatisfies(i)p{p(t,s,j)=0}=1whens>t;(1.2)(if)ac(tl,sl,jl)andp(tz,sa,j~)aremutuallyindependentwhenif/iZor81/s2;(ill)FOrfords,j,p(t,s,i)fort2sisanonhomogeneousMarkovChainwithtransitionprobabilityandinitialdistributiont=0T=0(2){Mt}isani.i.d.non-negativeinteger-valuedtimeseriesandindependentofn(t,sli).Inthefollowing,weassumeEMtz相似文献   

变系数四阶边值问题正解存在性

该文结合算子谱论,应用锥不动点定理,建立了四阶边值问题\[\left\{ {\begin{array}{l}u^{(4)} + B(t){u}' - A(t)u = f(t,u),0 < t < 1 ,\\u(0) = u(1) = {u}'(0) = {u}'(1) = 0 \end{array}} \right.\]正解存在性定理,这里$A(t),B(t) \in C[0,1]$,$f(t,u):[0,1]\times[0,\infty ) \to [0,\infty )$连续.     

MONOTONE ITERATIVE TECHNIQUE FOR NONLINEAR NEUTRAL DELAY DIFFERENTIAL EQUATIONS

1Introductiondescently,themonotoneiteratiyetechniqueissuccessfultoprovetheexistenceofextremalsolutionsofvariousnonlinearproblemforordinarydifferentialequations,delaydifferentialequations,integro-differentialequationsetc.,see[1--101.Inthispapertweshallconsidertheexistenceofextremalsolutionsoftheinitialvalueproblem(IVPforshort)fornonlinearneutraldelaydifferentialequationswherefEC[IOxRxRxR,R],CO=C[[--a,0],R],IO=[to,to T],to20,T>0,a<0,T<0,--a=adn{a,T},I=f--a,0]andforanyteIO,u(t s)ECO,sE…     

Γ-CONVERGENCE OF INTEGRAL FUNCTIONALS DEPENDING ON VECTOR-VALUED FUNCTIONS OVER PARABOLIC DOMAINS

51.IntroductionWebeginwiththecharacterizationofr--convergencein[1,2].Definition1.1.Let(X,T)beafirstcountabletOPologicalspaceand{F'}7=,bease-quenceOjfunctionalshemXtoR=RU{--co,co},u6X,AER.Wecallifandonlyifforeverysequence{u'}concealingtouin(X,T)andthereedestsasequence{u'}conveneingtouin(X,T)suchthatWecallA~r(T)timF"(u)ifandonlyifjoreveryah-- a(h-co)Throughoutthispaper,weassumethatfiisaboundedopensetinR".Letp>1,T>0,andmbeapositiveinteger.Denoteforavectorvaluedfunctionu.ConsiderthefUc…     

SELF-INTERSECTIONS OF BROWNIAN MOTIONON THE SIERPRISKI GASKET

1IntroductionIn[1],BarlowandPerkinsconstructeda"Brownianmotion"takingvaluesintheSierpinskigasket,afractalsubsetofR',thisisapointrecurrentsymmetric'diffosionprocesscharacterizedbylocalisotropyandhomogeneityproperties.Inthispaper,wediscussforwhichE,X(E)4{X(t),tEE}hask-multiplepoints,thatisdoesthereexistxsuchthatX(tl)=X(tz)=''=X(tk)=xforatleastkdifferenttieE?Thestructureofthispaperisasfollows.InSection2,somedefinitionsarerecalled.InSection3,wediscusstheealstenceofk-lllultiplepoints.Throu…     

THE OCCUPATION DENSITY FIELD FOR THE CATALYTIC SUPER-BROWNIAN MOTION

61.IntroductionLetW=[w,,ll.,.,s,t2o,aER']denoteastandardBrownianmotioninRdwithsendgroup{Pt,t2o},C(R')denotetheBanachspaceofcontinuousboundedfunctionsonRdequlppedwiththesupnorm.rk.(a):=(1 Ial')-p/',aERd,Cr(R'):-{feC(R'),lf(x)l5CIof.(x)}withsomeconstantCj,Mv(R'):={pisffedonmeasureonRdandJ(1 lxlp)-'p(dx)d.(P,i):=ff(x)p(dx),AisLebesgUemeasure.GiventheordinaryMP-valuedsuper-Brownianmotiono:=[o,,fl1,Ps,#,t2s2O,pEMP]asthecatalyticmedium,Dawsona…     

Numerical Analysis of Nonstationary Thermistor Problem

1.AMathematicalModelandaDiscreteSchemeThemodelofanonstationaxythermistorproblemisderivedfromtheconservationlawsofcurrentandenergy(see[l][2]l3]):Findapair{W,u}suchthatV-(a(u)VW)=oinQT=flX(O,T),(1.l)W=Woonoflx(O,T),(1.2)ut-bu=a(U)IVWl'inQT,(1'3)u=oonoflx(O,T),(1.5)u(x,o)=uo(x)infl(1.5)whereflCR"(N21)isaboundeddomain,occupiedbythethermistor;W=yt(x,t),u=u(x,t)aredistributionsoftheelectricalpotentialandthetemperatureinfl,respectively;J(u)isthetemperaturedependentelectricalconductivity;a…     

DMk OPERATORS AND SPECTRAL OPERATORS

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.；     

The fractal dimensions of the level sets of the generalized iterated Brownian motion

Let{W1(t), t∈R+} and {W2(t), t∈R+} be two independent Brownian motions with W1(0) = W2(0) = 0. {H (t) = W1(|W2(t)|), t ∈R+} is called a generalized iterated Brownian motion. In this paper, the Hausdorff dimension and packing dimension of the level sets {t ∈[0, T ], H(t) = x} are established for any 0 < T ≤ 1.     

算子半群母元豫解式的增长阶和它的指数稳定性

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.     

对2000年一道高考试题的研究与推广

今年全国高考数学理科第 (2 0 )题为 :( )已知数列 { cn} ,其中 cn =2 n + 3n,且数列 { cn+ 1 - pcn}为等比数列 ,求常数 p:( )设 { an}、{ bn}是公比不相等的两个等比数列 ,cn =an + bn,证明数列 { cn}不是等比数列 .这是一道“主要考查等比数列的概念和基本性质 ,推理和运算能力”的好题 .从本校许多考生的信息反馈来看 ,该试题起点低 ,入手宽 ,且具有一定的难度和较好的区分度 .经研究 ,笔者发现该试题所述的两个问题可归结为同一个模型 ,从而可用统一的方法加以解决 .定理　设 a、b、c、r、s、t均为实常数 ,则等式　　　 arn-1 + b sn-1 =c tn-1 (* )对任意的 n∈ N恒成立的充要条件为　　　　 a =b=c=0 ;(1)或　　 a + b=c=0 ,r=s;(2 )或　　 a =0 ,b =c,s=t;(3)或　　 b =0 ,a =c,r=t;(4 )或　　 a + b=c,r=s=t. (5 )证明　 (充分性 )逐一验证 (1)～ (5 )知它们均可分别使 (* )对任意的 n∈ N恒成立 ,故“充...     

从属过程的样本轨道的填充测度函数

设随机过程x(t)(t∈[0,∞))是一个从属过程,X[0,1]=|x∈R:t∈[0,1],X(t)=x}.通过对x(t)的指标函数g(u)(u≥0)的细致分析得到了关于X[0,1]填充测度函数的判别准则.事实上,若φ(s)(s∈(0,1))为任一测度函数,记h(s)=φ(s)/g(1/s),则有:h-p(x|0,1|= 相应于∫¹₀φ²(s)/s ds{<+∞ =+∞