一类五阶线性周期缓变系统周期解的存在唯一性

考虑5阶线性方程 x(5)+a,(t)戈(屯)+a:(t)x(3)+a:(t)x(2)+a‘(t)劣(‘)+as(t)x=e(t)将方程(1)化为等价方程组(1)一.、J,自-(勒dX_,,‘、。.,,。—=月、‘户了飞一I、‘夕dt这里X=(二,,…,戈5)’,A(t)=(a‘,(t)),f(t)=(o,o,o,o,e(t)),=a一。=1,aol二一a。,a。:=一a4,a。,=一a3,a。‘=一az,a。。=一al,,j=1,2,“·,5.我们得到如下的 定理.假设方程(1)满足如下条件 1 .a‘(t)连续可微,e(t)连续,且a‘(t+T)=a‘(t),e(t+T)=<月,{e(t)}相似文献   

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…     

SOME LIMINF RESULTS ON INCREMENTS OF THE PRIMITIVES OF BROWNIAN MOTION

§ 1　Introduction and ResultsLet{W(t) ;t≥ 0 }be a standard Brownian motion.For a positive integer m,define aGaussian processXm(t) =1m!∫t0 (t-s) md W(s) ,(1 .1 )which was firstmentioned by[1 ] .This class of processes arises in several domains of ap-plied mathematics.For instance,the process X1 (· ) ,which has been studied at length,isthe solution of the Langevin's equation under certain physical conditions.Wahba[2 ,3] usedXm(· ) to derive a correspondence between smoothing by splines …     

OSCILLATORY AND ASYMPTOTIC BEHAVIORS OF FIRST ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper the author discusses the following first order functional differentialequations: x'(t) +integral from n=a to b p(t, ξ)x[g(t, ξ)]dσ(ξ)=0, (1) x'(t) +integral from n=a to b f(t, ξ, x[g(t, ξ)])dσ(ξ)=0. (2)Some suffcient conditions of oscillation and nonoseillafion are obtained, and two asymptolioproperties and their criteria are given. These criferia are better than those in [1, 2], and canbe used to the following equations: x'(t) + sum from i=1 to n p_i(t)x[g_i(t)] =0, (3) x'(t) + sum from i=1 to n f_i(t, x[g_i(t)] =0. (4)     

THE GEOMETRY OF HIGHER PATH-SPACES

<正> IntroductionThe geometry in a space R_n(x~1,…, x~n)in whit.h there is givena system of differential equations of the r-th order:(?)has recently been studied by various writers.It seems to me thatthe most natural way to study this problem is to formulate the problemas a problem of equivalence.By solving complelely the problem ofequivalence the geometry in the space is automatically defined.By theuse of the general method of Cartan the problem of equivalence and thedefinition of a geometry trom a geometric object become two aspects ofthe same problem.We want to illustrate this point clearly in the (?)s-eussion of the present problem.     

单位球面上具有非负M(o)bius截面曲率的超曲面

Let x : M → Sn 1 be a hypersurface in the (n 1)-dimensional unit sphere Sn 1 without umbilic point. The M(o)bius invariants of x under the M(o)bius transformation group of Sn 1 are M(o)bius metric, M(o)bius form, M(o)bius second fundamental form and Blaschke tensor. In this paper, we prove the following theorem: Let x: M → Sn 1 (n ≥ 2) be an umbilic free hypersurface in Sn 1with nonnegative M(o)bius sectional curvature and with vanishing M(o)bius form. Then x is locally M(o)bius equivalent to one of the following hypersurfaces: (i) the torus Sk(a) × Sn-k(√1-a2) with 1 ≤ k ≤ n - 1; (ii) the pre-image of the stereographic projection of the standard cylinder Sk × Rn-k (∩) Rn 1 with 1 ≤ k ≤ n - 1; (iii) the pre-image of the stereographic projection of the cone in Rn 1: ～x(u, v, t) = (tu, tv),where (u,v,t) ∈ Sk(a) × Sn-k-1( √1- a2) × R .     

ON THE FUNCTIONAL-DIFFERENTIAL EQUATION OF THE DISTRIBUTION OF PRIME NUMBERS

1IntroductionsandStatementsInt-,iloIf)4-0'sCllorw'3I](see(ieVisme[4],Wright[:l]alldDriver[1])al,t}(}Inr)1,cdforil](lal[(lllrisl-,i(ital]yarlalyti(:itlI)rooftilt)prim(?number1,if(3or'3m5whi相似文献   

EXACT BOUNDARY CONTROLLABILITY OF 1-D NONLINEAR SCHRODINGER EQUATION

In this paper, the boundary control problem of a distributed parameter system described by the Schr(o)dinger equation posed on finite interval α≤ x ≤β:{iyt yxx |y|2y = 0,y(α,t) = h1(t),y(β,t) = h2(t) for t ＞ 0 (S)is considered. It is shown that by choosing appropriate control inputs (hj), (j = 1,2) one can always guide the system (S) from a given initial state ψ∈ Hs(α,β),(s ∈ R) to a terminal state ψ∈ Hs(α,β), in the time period [0, T]. The exact boundary controllability is obtained by considering a related initial value control problem of Schr(o)dinger equation posed on the whole line R. The discovered smoothing properties of Schr(o)dinger equation have played important roles in our approach; this may be the first step to prove the results on boundary controllability of (semi-linear) nonlinear Schr(o)dinger equation.     

THE EXISTENCE OF OVERLARGE SETS OF IDEMPOTENT QUASIGROUPS

A idempotent quasigroup (Q, o) of order n is equivalent to an n(n-1)×3 partial orthogonal array in which all of rows consist of 3 distinct elements. Let X be a (n+1)-set. Denote by T(n+1) the set of (n+1)n(n-1) ordered triples of X with the property that the 3 coordinates of each ordered triple are distinct. An overlarge set of idempotent quasigroups of order n is a partition of T(n+1) into n+1 n(n-1)×3 partial orthogonal arrays A_x, x∈X based on X\{x}. This article gives an almost complete solution of overlarge sets of idempotent quasigroups.     

NUMERICAL APPROXIMATIONS OF A SEMI-LINEAR ELLIPTIC PROBLEM

1IntroductionConsidertl1eparameterdependentequationwhereA,ltERarepara1ueterandf:R-RandS:R-RaresinoothoddfunctionwithLetS1:u(x)-u(T-x),r={S1,I}then(l.1)isr-equivariant.Tl1eequality(1.2a)isjustanormalizatiolloffatx=o.Otherwise,onelllayreseektl1eparameterxtoensure(1.2a).To8implifyanalysisweintroduceaSobolevspaceX:=Hl(o,1)anddefilleamappingT:gEL'(o,T)-lL:=TgEXimp1icitly:for(Tg)'V'dx=-jorgl)dx,VvEX,aweakformof(1.1)inXxR2isDuetof(o)=O,theproblem(1.3)(resp.(1))hasatrivialsolutioncurvesC…     

THE HAUSDORFF MEASURE FOR THE SAMPLE PATH OF THE WESTWATER PROCESS

91.IntroductionTheso-calledWestwaterprocessisthecoordinateprocessunderthepolymermeasureu(g)in3-dimensionsconstructedbyWestwater(see[7,8]),andthepolymermeasurev(g)onCo([O,1]-R')isformallydefinedbywhereC,isthenormalization,gE[o,oo)isacouplingconstant,pistheWienermeasureonCo([o,11-R').Inarecentpaper(see[9]),weprovedthattheHausdorffdimensionofthesamplepathoftheWestwaterprocess{X,,t6[O,1]}isalsotwowithprobability1tou(g).Actually,weprovedin[9]thatgh(ImX)5C,y(g)-a.e.(1.1)m(ImX)5C,y(g)-a.e.(1.…     

ON THE METHOD OF SOLUTION FOR A KIND OFNONLINEAR SINGULAR INTEGRAL EQUATION

The solutions of the nonlinear singular integral equation ψo(t)2 2b/πi ∫L ψ(τ)/T-t dr =f(t), t ∈ L, are considered, where L is a closed contour in the complex plane, b ≠- 0 is a constant and f(t) is a polynomial. It is an extension of the results obtained in [1] when f(t) is a constant. Certain special cases are illustrated.     

PROPERTIES IN THE LARGE OF QUADRATIC SYSTEMS IN THE PLANE

The author proves that any quadratic system in the plane can be changed into the quadratic system (E_1) or (E_2) by means of linear transformations, and then gives a necessary and sufficient condition for the systems (E_1) and (E_2) to be bounded for t≥0 and to have precisely one monotone unboundol orbit for t≥0 respectively.     

ON THE ERROR FUNCTION OF THE SQUARE-FULL INTEGERS

Let L(x) denote the number of square-full integers not exceeding x. It is proved in [1] thatL(x)～(ζ(3/2)/ζ(3))x~(1/2) (ζ(2/3)/ζ(2))x~(1/3) as x→∞,where ζ(s) denotes the Riemann zeta function. Let △(x) denote the error function in the asymptotic formula for L(x). It was shown by D. Suryanaryana~([2]) on the Riemann hypothesis (RH) that1/x integral from n=1 to x |△(t)|dt=O(x~(1/10 s))for every ε>0. In this paper the author proves the following asymptotic formula for the mean-value of △(x) under the assumption of R. H.integral from n=1 to T (△~2(t/t~(6/5))) dt～c log T,where c>0 is a constant.     

ON THE PLANE SECTIONS OF THE TANGENT SURFACE OF A SPACE CURVE

<正> 1.Introduction.Let the tangent surface of an analytic curve Cin ordinary space be cut by a plane through a tangent at an ordinarypoint O of C,then the plane section thus obtained has a point of in-flexion at O,provided that the plane is not osculating to C at O.Thisis a well-known fact.Prof.Bompiani has enriched the projectivedifferential geometry of a plane curve at the neighbourhood of a point ofinflexion by introducing various covariant osculants.Prof.Su hasgiven some remarkable properties of the loci of the cusp-tangent t ofthe seven-point cusped cubic of the plane section of the tangent surfacemade by a variable plane n through a tangent of C at O.Prof.Lanehas investigated the loci of the osculants and various points and linesintroduced by Prof.Bompiani.The object of this note is to obtain someproperties of other loci related to the same problem as considered byProf.Su.     

D-CONVERGENCE AND STABILITY OF A CLASS OF LINEAR MULTISTEP METHODS FOR NONLINEAR DDES

1. IntroductionConsider the following nonlinear delay problem{:;\f>>:v{t(tf,?,,<,>3,<'~">>,:: 5:,3Ti:,,,,, [l:::;where y: R - C",T > 0 is a delay term, f: [t.,T] x CN x CN - CN and W(t):[to -- T, tol - CN denotes a given initial function. Thoroughout this paper 9 the problem(1.1) is supposed to have a unique solution y(t), which satisfies11 y(')(t) 115 Mi, t e [to ~ T,T]here norm 11. 11 is defined by 11 x II'=< xgx > (Vx E C"), and Mi > 0 are someconstants.Definition 1.1.[1] The clas…     

THE DIRICHLET PROBLEM FOR DIFFUSION EQUATION

Let D be a bounded domain in the d 1-dimensional Euclidean space R~(d 1). This paper aims at giving a probabilistis treatment of the Dirichlet problem for the following diffusion equation on D(1/2⊿ q)u(x,t)=/(t)u(x,t),(x,t)∈D,where q is a function to be specified later and ⊿ is the Laplace operator sum from i=1 to d(~2/(x_i~2)). The existence and uniqueness theorems are given, and furthermore, the probabilistie representation and martingale charaeterization of the solutions for diffusion equations are obtained.     

AN EXPONENTIAL STABILITY PROBLEM ON THE SMALL DELAY LINEAR SYSTEM

1IntroductionLetXbea'Banachspace,andB(X)betheBanachspaceofcontinuouslinearoperatorsfromXintoX.LetT(t)bea(CO)selltigroup,andletAbeitsinfinitesilllalgenerator.WedenotethedomainandtherangeofAbyD(A)andR(A).LetBbelongtoB(X).FOrthelinearsystem:andthedelaylinearsystem:wherer>0,((.)EC([--.,o],x)={fif:[--r,oj~X,fiscontinuous},r(t)EC[o,.)'05r(t)5r.Weconsidertileexponentialstabilityequivalencebetweellthesolutionof(1)and(2),whichllleans:ifthereexistM,a>0,sllththatIIS(f)II5Me--"'(f20),whereS(…     

THE BLOW-UP PROPERLIES OF SOLUTIONS TO SEMILINEAR HEAT EQUATIONS WITH NEUMANN BOUNDARY CONDITIONS

1IntroductionInthispaper,weconsidertheinitial-boundaryvalueproblem,where"o(x)isnon-negativesmoothfunctionsatisfying"o.(0)=0,"o.(1)=1'Whenconsideringtheblow-upofsolution,thefollowingproblemarisenaturally:Doesblow-upoccur?Howdoesthesolutionapproachtheblow-uptime'!Andwilersisthehotspotlocated(blow-upset)?WelookattheheatequationwithanonlillearboundaryconditiollHerefiisaboundeddomaininR",p>1isarealnumber.IthasbeenknownforalongtimethattheDroblelil(1.1).(1.2)withAL(~,0)--'no(x)doesnothaveaglobals…     

OSCILLATION CRITERIA FOR SECOND ORDER QUASILINEAR DIFFERENTIAL EQUATIONS

1 IntroductionThis paper is concerned with the problem of oscillation of second-orderquasilinear differential equationsFOr equation (1), the following conditions, which are referred to (FO), arealways assumed to be valid(a) a > 0 is a constant;(b) F: [to, co) x R x R x R x R - R is a continuous function;(c) sgn F(t, x) u? v3 w) = sgn x for each t 2 to and x, u? v? w E R,(d) r(t) E C([to, co); (0, co)) and jco ddt = co.' (e) T: [to, co) - R is continuous, T(t) 5 t for t 2 to and ill7T(t)…