VISCOSITY SOLUTION THEORY OF A CLASS OF NONLINEAR DEGENERATE PARABOLIC EQUATIONS Ⅱ. LIPSCHITZ CONTINUITY OF FREE BOUNDARY

0.IntroductionIn[1],weestablishthecomparisonprinciple,regularity,existenceoftheviscositysolutionsforthefollowingfullynonlineardegenerateequations(Problem(C))'withFsatisfyingrespectivelysomeofthefollowingassumptions:F(q,X)eC(R"xS"),(FI)Fisdegenerateparabolic,i.e.F(q,X Y)SF(q,X),VY30,whereSadenotesthespaceofnxnsymmetricmatriceswiththeusualordering.RevisedJune2,1994.*ThisresearchissupportedbytheNationalNaturalSciencesFoundationofChina.whereifXlll~ltrX l ltrX--I,X=X Xo,X 30,X--50,A,…     

齐型空间上的一些新的Triebel-Lizorkin空间及其标架特征

设(X,ρ,μ)d,θ是齐型空间,ε∈(0,θ],|s|＜ε且max{d/(d+ε),d/(d+s+ε)}＜q≤∞.引进了一类新的Triebel-Lizorkin空间Fs∞q(X),并通过先建立与空间Fs∞q(X)的范数相关的Plancherel-Polya型不等式的方法建立了这些空间的标架特征;给出了当|s|＜ε,max{d/(d+ε),d/(d+s+ε)}＜p≤∞且0＜q≤∞时,Besov空间Bspq(X),以及当|s|＜ε,max{d/(d+ε),d/(d+s+ε)}＜p＜∞且max{d/(d+ε),d/(d+s+ε)}＜q≤∞时,Triebel-Lizorkin空间(F)spq(X)的标架特征;此外,还引进了与给定仿增函数b相关的新的Triebel-Lizorkin空间b(F)s∞q(X)和H(F)s∞q(X),并且建立了空间b(F)s∞q(X)和空间H(F)s∞q(X)的相互关系;进一步证明了如果s=0且q=2,则H(F)s∞q(X)=(F)s∞q(X).因为(F)s∞q(X)=BMO(X),所以事实上这也给出了空间BMO(X)一个新的特征刻画.     

APPROXIMATE CONTROLLABILITY OF INFINITE DIMENSIONAL BILINEAR SYSTEMS

1.IntroductionRecentlymanyresultshavebeenobtainedfordistributedbilinearsystemsl3'7'8].In[3]thecontrollabilityofabilinearsystem%=Aw(t) p(t)Bw(t)wasstudied,whereAistheindnitesimalgeneratorofaCosemigroupofboundedlinearoperatorsT(t)onaBanachspaceX.B:X-- 5isaboundedlinearoperator,andpELI([0,T];R)isacontrol.TheconditionsforelementsofEtobeaccessiblefromagiveninitialstatecoweregiven.Itisclearthattheystudiedthebilinearsysteminaspecialcase.In[8]theystudiedthesysteminamorespecialcasebecausetheyass…     

Maximal regularity of second order delay equations in Banach spaces

We give necessary and sufficient conditions of Lp-maximal regularity(resp.B sp ,q-maximal regularity or F sp ,q-maximal regularity) for the second order delay equations:u″(t)=Au(t) + Gu't + F u t + f(t), t ∈ [0, 2π] with periodic boundary conditions u(0)=u(2π), u′(0)=u′(2π), where A is a closed operator in a Banach space X,F and G are delay operators on Lp([-2π, 0];X)(resp.Bsp ,q([2π, 0];X) or Fsp,q([-2π, 0;X])).     

ON THE UPPER BOUND OF THE NUMBER OF PRINES IN ARITHMETIC PROGRESSION

Let a and q be relatively prime positive integers and π(x;q,a)stand for the number ofprimes p≤x congruent to a and q.H.Iwanice proved thatπ(x;q,a)<((2 ε)x)/(φ(q)log D)(1)for any ε>0,x>x_0(ε)and q≤x~(9/20(?)),where D=x q-~(3/8).The author applies an improved estimation of the error term in the linear sieve,provesthat for any ε>0,x>x_0(ε)and q≥x~(5/11-ε),(1)is true.     

ON THE UNIFORM CONSISTENCY OF THEKAPLAN-MEIER ESTIMATOR UNDER HEAVY CENSORING

Let X,i.i.d. and Y1i. i.d. be two sequences of random variables with unknown distribution functions F(x) and G(y) respectively. X, are censored by Y1. In this paper we study the uniform consistency of the Kaplan-Meier estimator under the case ey=sup(t:F(t)＜1)＞to=sup(t2G(t)＜1) The sufficient condition is discussed.     

Englixh An Interior Point Algorithm forthe Parametric Linear Complementarity Problem

1.IntroductionLetLCP[t]:S~MX q tq,S20,X20,STX~0bealinearcomplementarityproblem(LCP)withtERasitsparameter.Inthispaper,theproblemisassumedtobemonotone,i.e.,MEboxsispositivesemi-definite.Itisalsoassumedthattheproblemisnon-trivial,i.e.,thereexistsatERsuc...     

一些集的基数(Ⅸ)

Let X be an infinite set, C={G:G is a group defined on the X}, Define {H:H is isomorphic to G}, C_2={C & G js not commutative},then |C_2|=2 If K={F:F is a division ring defined on the X},K_1={K & F is not com mutative}, K_2={K & F is commutative},then |K_1| [ = IK2t --2:s~. Suppose T(X) {X~X & qψis bijective},S={G:G is a subgroup of T(X)},S_1={S & G is commutative}, S_2={S & G is not commutative},then |S_1|=|S_2|=2~(2~(|x|)).     

两类特殊二元一次方程组的解

二元一次方程组{(1)当a:、b:、a一x b.夕=cla:x bZ少=cZc:和a:、b:、(!忿;会;},。)c:分别成等差数列时,方程组的解是{(2)当a:、b:、劣=一1y二2c;和a:、bZ、。2分别成等比数列且公比分别为q:、q,时,方程组的解是{y=证明:(l)一q一qZq一 q:将方程组改写成a:‘ (a: d:)夕=a: Zd;aZ二 (a: dZ)夕=a: Zd:(I)(I)(a:b:一匕:d:斗。)(I)xa:一(I)xa:,得(a;d:一a,dZ)夕=2(a Zd:一a,dZ)(2) 夕=2代入(I)〔或(I)〕得x=一1.将方程组改写成.’.广“一1 、夕=2。X q lyx q:y=好=q量(I)(F)(g:一Q:车。)(l(一(F),得(g:一g;)少=g荃.’.y=q: qZ,代入(l)一…     

Some Notes on a Minimization Problem

Let X[a,b] be a compact set containing at least n+1 points and Kan n-dimensional Haar subspace in c[a,b]. Let F(x,y) be a nonnegativefunction, defined on X×(-∞,∞), satisfying ‖F(·,p)‖<∞ with the L_∞norm forsome∈K, where F(x,p)≡F(x,p(x)). The minimization problem discussed in this paper is to find an elementp∈K such that ‖F(·,p)‖=inf ‖F(·,q)‖, such an element p(if any) is saidto be a minimum to F in K~(q∈K). The author in [1,2] studied this problem and has given the main theoremsin the Cbebyshev theory under the following assumptions: (A) lim F(x,y)=∞, x∈X; (B) lim F(x,u)=F(x,y), x∈X,y; (C)lim F(u,υ)=F(x,y),x∈X,y; (D) For each x∈X there existtwo real numbers f~-(x) and f~+(x),f~-(x)f~+(x). such that F(x,y) is strictlydecreasing with respect to y on (-∞,f~-(x)] and strictly increasing on [f~+(x),∞), and F(x,y)=F(x):=inf F(x,υ) on [f~-(x),f~+(x)]. Denote f_1(x)=inf{y:F(x,y)‖F~*‖},f_2(x)=sup{y:F(x,) ‖F‖},f_1(x)=lim f_1(u),f_2(x)=lim f_2(u), G=(q∈K: f_1qf_2}.For pεK set X_p={     

Integral Operators Between Fock Spaces∗

In this paper, the authors study the integral operator Sφf(z) = Z C φ(z, w)f(w)dλα(w) induced by a kernel function φ(z, ·) ∈ F ∞α between Fock spaces. For 1 ≤ p ≤ ∞, they prove that Sφ : F 1 α → F p α is bounded if and only if sup a∈C kSφkakp,α < ∞, (?) where ka is the normalized reproducing kernel of F 2 α; and, Sφ : F 1 α → F p α is compact if and only if lim |a|→∞ kSφkakp,α = 0. When 1 < q ≤ ∞, it is also proved that the condition (?) is not sufficient for boundedness of Sφ : F q α → F p α . In the particular case φ(z, w) = eαzw?(z ? w) with ? ∈ F 2 α, for 1 ≤ q < p < ∞, they show that Sφ : F p α → F q α is bounded if and only if ? = 0; for 1 < p ≤ q < ∞, they give sufficient conditions for the boundedness or compactness of the operator Sφ : F p α → F q α.     

ON CLOSED ORBIT FOR A CLASS OF PLANAR SYSTEMS OF HIGH DEGREE

1.IntroductionThepolynomialdifferentialequationofdegreep qwherep2l,q21areintegersandx20,y20,a>0arerealnumber,asknownin[12],describesthedynamicalbehaviorsofmulti-molecularbiochemicalreactionHerex,yandp,qdenoterespectivelytheconcentrationsofsubstancesandnumbersofmolecules.UsuallyEq.(1.1)iscalled"TI-moleculesmodel"withn~p q.LikePrigogine'sB.....l.t.rI4],Higgins'm.del[5]andTyson-Lighe'smodel[6],itisofsignificanceinourunderstandingofthephenomenaofnonequilibriumstates[3].Thecaseofn=3isdiscussedp…     

森谱的界

设λ_k(F)是树或者森的第k大特征值,[x]是不超过x的最大整数,q是F的边独立数.本文证明了:对于1≤k≤[(q 1)/2]有λ_k(F)≥1,并且这个下界是最好可能的;对于1≤i≤[q/2],若q为偶数,则有λ[(q 1)/2] i(F)≥2cos((2iπ)/(4i 1)),若q为奇数,则有λ_([(q 1)/2] i)(F)≥2cos(((2i 1)π)/(4i 3)),     

RELATIONS BETWEEN PACKING PREMEASURE AND MEASURE ON METRIC SPACE

Let X be a metric space andμa finite Borel measure on X. Let pμq,t and pμq,t be the packing premeasure and the packing measure on X, respectively, defined by the gauge (μB(x,r))q(2r)t, where q, t∈R. For any compact set E of finite packing premeasure the authors prove: (1) if q≤0 then pμq,t(E)=pμq,t(E);(2)if q>0 andμis doubling on E then pμq,t(E) and pμq,t(E) are both zero or neither.     

单调空间与度量化定理

本文回答了关于MCM空间遗传性的一个问题,讨论了k-MCM空间是k半层空间的条件,得到了一些用g函数刻划的度量化定理.主要结论有:MCM空间是关于Fσ子空间遗传的;在正规空间类中,q空间(ωN空间,k-MCM空间)是关于开Fσ子空间遗传的;如果X是具有Gδ对角线的正则次中紧 k-MCM空间,则X是k半层空间;X是可度量化空间的充要条件是存在X上的g函数满足对X中任意不相交的闭集F与紧集C,都有某个n∈ω,使得(∪x∈F g(n,x))∩(∪y∈C g(n,y))=(?).     

A KIND OF IMPLICIT ITERATIVE METHODS FOR ILL-POSED OPERATOR EQUATIONS

1.IntroductionLetX,YbetworealHilbertspacesandletA:X-- Ybeaboundedlinearoperator.ConsidertheoperatorequationAx~y.(1.1)IfR(A),i.e.,therangeofA,isnonclosedinY,equation(1.1)isill-posed[1].Manyimportantproblemsinappliedsciencesresultinthiskindofequations...     

A FORMULA FOR INDEX OF A SINGULAR POINT OF POLYNOMIAL DIFFERENTIAL SYSTEMS IN THE PLANE

1IntroductionandConclllsionIntheplallalrPolynomialdifferentialsystem$~p.(x,y) of(x,y)=X(x,y),Z~q.(x,Y) '()(x,y)~I'(x,y)(l)wherep.(x,y),q.(x,y)arehomogenouspolynomialsinigyofthedegreen>1,(x,y)ER',withq:(x,7/) p:(x,y)/0,andof(x,y),,of(x,]j)arepolynomialinx,…     

ON THE PROBLEM OF NECESSARY CONDITIONS ENSURING UNIFORM CONVERGENCE OF KERNEL DENSITY ESTIMATES

Let X_1,…,X,be a sequence of p-dimensional iid.random vectors with a commondistribution F(x).Denote the kernel estimate of the probability density of F(if it exists)by_n(x)=n~(-1)h~_n(-p)K((x-X_i)/h_n)Suppose that there exists a measurable function g(x)and h_n>0,h_n→0 such thatlim sup丨f_n(x)-g(x)丨=0 a.s.Does F(x)have a uniformly continuous density function f(x)and f(x)=g(x)?This paperdeals with the problem and gives a sufficient and necessary condition for generalp-dimensional case.     

THE JUMPING PHENOMENON OF THE DIMENSIONS OF COHOMOLOGY GROUPS OF TANGENT SHEAF

Let X be a compact complex manifold.Consider a small deformation π :X→B of X,the dimensions of the cohomology groups of tangent sheaf H q(X_t,T_X_t) may vary under this deformation.This article studies such phenomena by studying the obstructions to deform a class in H q(X,T X) with parameter t and gets a formula for the obstructions.     

STRONG REPRESENTATIONS OF THE SURVIVAL FUNCTION ESTIMATOR ON INCREASING SETS FOR TRUNCATED AND CENSORED DATA

1991MRSubjectClassification62G05,62E201IntroductionandtheMainResultsInthisarticlewestudysllrvivaldatawhicharesubjecttobothlefttrull(:ationandrightcensoring(LTRC).Morespecifically,let(X,T,Y)denoterandomvariableswhereXisthevariableofinterest,calledthelifetimevariable,withunknowndistributionfullctioll(d.f.)F;Tistherandomlefttruncatiolltiliiewitharbitraryd.f.G,andYistherandorl.1rightcensoringtirliewitharbitraryd.f.H.ItisassumedthatX,T,Yarerxlutuallyindependent.Intheran相似文献