Abstract. Taking an open and bounded interval in R,#

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convection-diffusion

1 Introduction
Mathematical models in physics and engineering often lead to di erential equations with coe cients that may di er by several orders of magnitude. Such problems can be found when modeling processes in chemical kinetics, transport of heat and mass, plate bending etc. Important characteristic of such problems is that the solution are highly localized by exhibiting boundary and internal layers, point and line singularities etc. The aim of a numerical technique for solving such problems is to nd a mesh which resolves these localized phenomena. This in turn is related to the question how accurately (and inexpensively) one can obtain information about the solution. A reasonable approach should include both a priori analysis of the problem and its solution and a posteriori analysis of the computational results in order to verify their accuracy and subsequently improve the results by re ning the mesh. The a priori analysis can be used to a priori construct the mesh. For the state-of-the-art research in this direction we refer to the monographs of Miller, O'Riordan, and Shishkin 13], H.-O. Ross, M. Stynes, and L. Tobiska 16]. The a posteriori analysis is used to construct the \best" mesh for the solution of a particular problem within given tolerance for the error. Practically, this means that starting with a very coarse initial mesh, further in the solution process, the mesh is re ned in a fully adaptive way, namely new grid points are added in the areas where the a posteriori error estimators and indicators suggest. For studies in this direction we refer to the monographs of I. Babuska, O. C. Zienkiewicz 1] and R. Verfurth 19]. The aim of the present paper is to derive and study unconditionally stable approximations of singularly perturbed problems of second order based on least-squares nite element method. Using

Dirichlet boundary control of semilinear parabolic equation

126
N. Arada and J.-P. Raymond
initial condition belongs to Wad ⊂ L ∞ ( ), and A is a second-order elliptic operator of the form Ay(x ) = − iN , j =1 Di (aij ( x ) D j y ( x )) (where Di denotes the partial derivate with respect to xi ). We are interested in optimality conditions in the form of Pontryagin’s principles for the control problem (P) where J ( y , u , v, w) =
Linear-quadratic problems of the form ( P ) (when (x , t , y , u ) is of the form a (x , t ) y + u and J is quadratic) have been studied for 30 years [18], [14]. More recently the Dynamic Programming approach has been developed for Dirichlet boundary controls of nonlinear parabolic equations [5], [4], [12]. In this approach the state variable is a time-dependent function with values in a Hilbert space. Very recently, growing interest has been taken in the corresponding problems with pointwise state constraints [6], [9], [10], [17], [20], [21]. For such problems, it is natural to look for solutions in L ∞ ( Q ) or in Cb ( Q ) (the space of bounded continuous functions in Q ). The main purpose of Part 1 is to develop new tools to analyze the state equation when solutions are bounded and continuous in Q , and next to obtain Pontryagin’s principles. The corresponding problem with pointwise state constraints is studied in Part 2. Taking advantage of the fact that v belong to L ∞ ( ), we prove existence and regularity results for (1) by a method different from the ones previously mentioned. We deﬁne the solution y of (1) by the transposition method and we prove that it is the limit of a sequence of solutions for equations with a Robin boundary condition (Theorem 3.9). This kind of property is well known for solutions of linear hyperbolic equations [15], [19], or linear elliptic equations [7]. Here the novelty is that we use the same method for nonlinear equations to prove that solutions belong to Cb ( Q ∪ T ). Moreover, following [16], we think that this method may also be interesting for numerical approximations. We prove that optimal solutions satisfy three decoupled Pontryagin’s principles, one for the distributed control, one for the boundary control, and the last one for the control in the initial condition (Theorem 2.1). These optimality conditions are obtained with a method of diffuse perturbations. Expansions for diffuse perturbations of a boundary | d (x , ) > τ }× ]τ, T [ (see control are obtained only in subcylinders Q τ = {x ∈ Theorem 5.2). Such tools will be very useful to study the corresponding problems with pointwise state constraints in Part 2 [2]. The plan of the paper is as follows. Assumptions and the main result (Theorem 2.1) are stated in Section 2. The state equation, the linearized state equation, and estimates needed for the proof of Theorem 2.1 are studied in Section 3. We recall some results for the adjoint equation in Section 4. Taylor expansions for diffuse perturbations are obtained in Section 5 and the proof of Theorem 2.1 is given in Secttions and Main Result

Hausdorff Convergence and Universal Covers

Hausdor Convergence and Universal Covers
Christina Sormani
Abstract
Guofang Weiy
We prove that if Y is the Gromov-Hausdor limit of a sequence of compact manifolds, Min , with a uniform lower bound on Ricci curvature and a uniform upper bound on diameter, then Y has a universal cover. We then show that, for i su ciently large, the fundamental group of Mi has a surjective homeomorphism onto the group of deck transforms of Y . Finally, in the non-collapsed case where the Mi have an additional uniform lower bound on volume, we prove that the kernels of these surjective maps are nite with a uniform bound on their cardinality. A number of theorems are also proven concerning the limits of covering spaces and their deck transforms when the Mi are only assumed to be compact length spaces with a uniform upper bound on diameter.

超越种族：莫里森新作《慈悲》中的“奴役”解析

诺贝尔获奖作家研究超越种族:莫里森新作《慈悲》中的“奴役”解析王守仁吴新云内容提要:莫里森新作《慈悲》描写了北美殖民地初期蓄奴制对黑人、印地安土著和白人契约劳工等各类奴隶的伤害,以及白人自由人如何为心灵或历史枷锁所累的情况。

本文通过对小说主要人物进行分析,深入探讨“奴役”的本质,认为莫里森“超越种族”的视角彰显了她对历史、社会和人心的深刻洞察。

关键词:托妮・莫里森种族蓄奴制奴役美国黑人文学作者简介:王守仁,南京大学外国语学院教授,主要从事英美小说研究。

吴新云,中央财经大学外国语学院副教授,主要从事美国黑人文学与文化研究。

T itle:Transcending Race:An Analysis of“Enslave ment”in T oniMorris onπs A M ercyABSTRACT:A M ercy,T oniMorris onπs ne w novel,describes the physical and s p ir2 itual enslave ment suffered by both slaves of different races(blacks,native I ndians,and white contracted laborers)under slavery and the“free”whites whose hist orical and mentalli m itati ons shaped their lives in the early col onial peri od of the Ne w Continent.Based on acl ose exa m inati on of the main characters in the novel,this paper discusses the essence of enslave ment,and argues that Morris onπs thinking transcends race and A M ercy shows her insight int o hist ory,s ociety and the hu man heart.Keywords:ToniMorris on,race,slavery,enslave ment,Afr o2American literatureAuthors:W ang Shouren<sr wang@>is a p r ofess or of English at Nanjing University,Nanjing,China(210093).H is research area mainly covers B ritish and American ficti on.W u X i n yun<subtle wind@>is an ass ociate p r ofess or at Central University of Finance and Econom ics,Beijing,China(100081).Her research f o2 cuses on Afr o2American literature and culture.1993年诺贝尔文学奖得主美国黑人女作家托妮・莫里森(Toni Morris on1931—)是当今当代外国文学　No.2,2009时代最杰出的小说家之一,她的小说成就标志着20世纪美国黑人文学史上继赖特、艾里森之后的又一座高峰。

Navier--Stokes

A fully parallel mortarﬁnite element projectionmethod for the solution of the unsteadyNavier–Stokes equationsA.BEN ABDALLAH and J.-L.GUERMONDAbstract.This paper describes the parallel implementationof a mortarﬁnite-element projection method to compute in-compressible viscousﬂows.The basic idea in the derivation ofthis method is that the appropriate functional setting for pro-jection methods must accommodate two different spaces forrepresenting the two velocityﬁelds calculated in the viscousand incompressible half steps of the method.The velocity cal-culated in the viscous step is chosen in the space constrainedby the mortar elements;that is,a weak continuity through thesubdomain interfaces is enforced,whereas in the projectionstep,the weak continuity is relaxed.As a result,the projec-tion step is fully parallel.The numerical solutions of a seriesof test problems in two dimensions calculated by the pro-posed method compare quite satisfactorily with the referencesolutions.1INTRODUCTIONThe projection method of Chorin[4,5]and Temam[14](seealso[13]and[12])is the most frequently employed techniquefor the numerical solution of the primitive variable Navier–Stokes equations.This method is based on a peculiar time-discretization of the equations governing viscous incompress-ibleﬂows,in which the viscosity and the incompressibility oftheﬂuid are dealt within two separate steps.A functional analytical setting which properly accounts for thedifferent character of the equations of the two half-steps hasbeen proposed recently by Guermond[8,9]and Guermond-Quartapelle[10].The aim of this work is to describe the par-allel implementation of aﬁnite-element projection methodwhich fully exploits the different mathematical structure ofthe two half-steps.The spatial discretization is based on amortarﬁnite-element approximation.In theﬁrst step,the vis-cous velocity is approximated by means of the mortarﬁnite element technique,whereas in the second step the constraint(3) In the sequel we assume that,are smooth solutions of the problem above and that at the initial time all the compatibility conditions implied by the required smoothness are satisﬁed. For instance,such conditions are satisﬁed if the initial datum is zero and the source term is regularized at.c1996A.Ben Abdallah and J.-L.Guermond ECCOMAS96.Published in1996by John Wiley&Sons,Ltd.2.2The spatial discretizationThe mortar element technique have been developed by Bernardi, Maday and Patera[1],[2].In this section we recall some as-pects of this technique and we formulate our time dependent Stokes problem within this discrete setting.We assume that we have at hand a partition of intonon-overlapping polygonals:and(4) For sake of simplicity we assume also that is two-dimensional, though the fractional step technique we develop is dimension independent.We restrict ourself to polygonal sub-domains. Furthermore,we assume that the decomposition of is ge-ometrically conformal;that is to say,the intersection of two subdomains is an edge,a vertex or empty.This hypothesis simpliﬁes the implementation of the technique but is not a limitation of the mortar method.The interface between sub-domain and is denoted by.For each subdomain,we deﬁne a regular mixed tri-angulation(e.g.-iso-or see Girault-Raviart [6]or Brezzi[3]for other details).We denote by andthe linear spaces spanned by the velocity and the pressure triangulation of respectively.Note that we do not require the grids of each subdomains to match;the weak continu-ity through the subdomains’interfaces is enforced by mortar functions.We denote by the space of the mortar func-tions associated to the interface;for a clear deﬁnition of this space we refer to[1].The space of the mortar functions is deﬁned by(5) For a function in we denote by the components of in.We now deﬁne,,the subspace of so that(6) where denotes the trace operator from onto .The space is equipped with the followingscalar product:(7) We also introduce the scalar product so that(8)We use this scalar product to deﬁne the dual norm of;the dual of equipped with the dual norm is hereafter denoted by.Finally,we deﬁne the projection onto.The pressure will be approximated in so that(9)is equipped with the scalar product(10) Let us also introduce the continuous bilinear formI R,so that for all in,(11) It can be shown that is coercive with respect to the norm of.(12) We associate with the linear continuous operatorso that,for all in,we have.We now introduce the continuous bilinear formI R so that(13) We associate with the continuous linear operatorand its transpose so that for every couple in we haveand.It can be shown that is onto;that is to say,there is a constant (independent of)so that(14) In the functional framework deﬁned above,the spatially dis-cretized time-dependent Stokes problem can be reformulated as follows.For andﬁnd and so that:The problem(15)can be shown to be well posed.We hereafter assume that the solution of this semi-discretized problem con-verges in the appropriate sense to that of(3);the convergence analysis is very classical.In the following we are interested only in approximating the time-dependent problem by means of a projection technique.3THE FRACTIONAL-STEPPROJECTION ALGORITHMS3.1The discrete settingIn order to uncouple the incompressibility constraint from the time evolution problem,we are led to introduce additional tools(see[8][9]for other details).We deﬁne theﬁnite dimensional linear space so that(1) We equip with the norm of and we denote this norm by.It is clear that is a subspaceof(in terms of linear space)and we denote by the continuous injection of into.Actually,is composed of the functions of which are weakly continuous across the subdomains’interfaces.We introduce another discrete version of the divergence op-erator;let be so that(2) The relation between and is brought to light by Proposition1is an extension of,and.A consequence of this proposition is that is also nec-essarily onto,for is onto.As a consequence,if we set ,we have a discrete counterpart of the classical decomposition:Corollary1We have the orthogonal decomposition:(3) 3.2The projection schemeWe introduce a partition of the time interval:for where,and deﬁne two series of approximate velocities and and one series of approximate pressures so that(5)The series is initialized by and assuming that the series is initialized by .Remark3.1.The problem(4)is well posed since is-elliptic.The problem(5)is also well posed thanks to corollary 1:indeed the couple is the decompo-sition of in;i.e.,where is the orthogonal projection of onto. Remark3.2.Note that since no weak continuity through the subdomains’interfaces is enforced on and,the projection problem(5)reduces to a series of completely independent problems that can be solved in parallel. Remark3.3.In practice the projected velocity is eliminated from the algorithm as follows(see[8]).Replace in(4)by its deﬁnition which is given by(5)at the-th time step;note that ,as already mentioned.In(5),is eliminated by applying to theﬁrst equation and by noting thatis an extension of.The algorithm which is implemented reads,for,(7) Remark3.4.Higher accuracy in time can be obtained if we replace the two-level backward Euler step ofﬁrst order by a backward three-level Euler step of second order as follows(9) Of course,the algorithm can be implemented in a more conve-nient form by eliminating the end-of-step velocity,as follows:(11) It can be shown that for aﬁxed mesh size,this algorithm yields second order accuracy in time in the natural norms deﬁned below.3.3Stability and convergence analysisThe projection algorithm introduced above is stable in the following sense:(12)Section Title3 A.Ben Abdallah and J.-L.Guermondwhere is a function of the data of the problem. The fact that the projection step is parallel is paid by the fact that the discrete gradient operator is not optimally stable; indeed if we denote by an interpolation operator on, we only haveof type(3) where is a mass matrix.This system is solved iteratively by means of a conjugate gradient preconditioned bywhere is the lumped mass matrix.4.2ImplementationThe code is written in FORTRAN90and uses the MPI message passing library.It is run on a CRAY-T3D which has128Dec Alpha processors.Each processor can deliverbut,still now,because of hardware imple-mentation(mutilated memory caches),hardly of the peak can be used.(recall that T3D is an ex-perimental machine,hopefully,these problems will be solved with the T3E).4.3Test problemsIn order to illustrate the second order algorithm described above,(10)-(11),we provide convergence tests on a test prob-lem.We consider the following exact solution on the square ,We have chosen linear functions in space to avoid spending much time on error calculation.The domain is divided into sixteen subdomains as shown inﬁgure2.Fore sake of simplicity,we used structured grid on subdomains.We solve the time-dependent Stokes problem on the time interval with a source term corresponding to our chosen solution.Inﬁgure3,we have reported the errors on the velocity in the norms and as functions of the time step for different global mesh sizes.As expected we observe a second order slope for moderate but we can see that the estimation constant seems to be dependent of in the rate of.This result is not surprising since we have proven convergence of order for theﬁrst order scheme(6)-(7);hence,convergence in time of orderfor(10)-(11)seems reasonable.(but has yet to be proven). The error on pressure in the norm of is re-ported as a function of inﬁgure4.The same second order slope and mesh size dependency as that obtained on the ve-locity are observed on the pressure.To illustrate the mesh size inﬂuence on the convergence, we have reported inﬁgure5the errors on both velocity and pressure as functions of while is kept equal to .We note a second order slope on the velocity in the norm.First order slope is obtained in thenorm(practically we obtained better than ﬁrst order for the term in is still dominant).The same observations can be made for the pressure.δt10−310−210−1100101ErrorsonUSTOKES ProblemErrors on the velocity UFigure3.andversus.δt10−310−210−1100101ErrorsonpSTOKES ProblemErrors on the pressure pFigure4.versus.Section Title5 A.Ben Abdallah and J.-L.Guermondh10−310−210−110E r r o r sSTOKES Problem( δt = c h 3/2, c=0.8192 )Figure 5.,andversuswith.4.4Speed-UpThe numerical tests reported in this section were performedwith different numbers of processors for a global mesh size(nodes)and with .In table1,we can see for different numbersof processors the elapse time for one time step iteration,the mono-processor equivalent time ,the elapse time and the number of Conjugate Gradient iterations (resp.the time and the number of Preconditioned Conjugate Gradient iterations)for one prediction (resp.projection)step.At ﬁrstsight,we can notice that the algorithm has almost the right speed-up (from to we have a globalspeed-up of).But if we look at the projection step we notice better results;recall that this step amounts to solve a Poisson problem.If we look at the prediction step,where the communications between processors occur,the speed-up be-tween and (resp.and )is about(resp.).Actually,if we consider the speed-up for one CG iteration of the prediction step we obtain (resp.).There are two reasons for this.First,in the prediction step,the linear system is solved by means of a Conjugate Gradi-ent algorithm which is not preconditioned.As a result,the total number of iterations that is needed to reach convergence is dependent of both the problem size and (the conditionnumber ofdepends on ).The second reason is that the cost of one CG iteration is not of order but rather between and (we use sparse matrix techniques,see forinstance [7]).The efﬁciency rate has not been calculated since the problem is too big to be solved on one single processor ().Table 1.Time inventory for one time step iteration.166.198.4 4.2765 1.88932 3.2102.3 2.36820.838641.7111.51.35940.398[9]J.-L.Guermond,Sur l’approximation des e´quations de Navier–Stokes instationnaires par une me´thode de projection,C.R.Acad.Sc.Paris,Se´rie I,319,1994,887–892.[10]J.-L.Guermond and L.Quartapelle,On the approximation ofthe unsteady Navier–Stokes equations byﬁnite element projec-tion methods,LIMSI report95-14,submitted to Numer.Math.[11]O.Pironneau,Me´thode des´le´mentsﬁnis pour lesﬂuides,RMA7,Masson,1988.[12]L.Quartapelle,Numerical Solution of the IncompressibleNavier–Stokes Equations,ISNM113,Birkha¨user,Basel,1993.[13]R.Temam,Navier–Stokes Equations,Studies in Mathematicsand its Applications,2,North-Holland,1977.[14]R.Temam,Une me´thode d’approximation de la solution dese´quations de Navier–Stokes,Bull.Soc.Math.France,98,1968,115–152.Section Title7 A.Ben Abdallah and J.-L.Guermond。

Vitali’s theorem and WWKL

VITALI’S THEOREM AND WWKLDOUGLAS K.BROWNMARIAGNESE GIUSTOSTEPHEN G.SIMPSONAbstract.Continuing the investigations of X.Yu and others,westudy the role of set existence axioms in classical Lebesgue mea-sure theory.We show that pairwise disjoint countable additivityfor open sets of reals is provable in RCA0.We show that sev-eral well-known measure-theoretic propositions including the VitaliCovering Theorem are equivalent to WWKL over RCA0.1.IntroductionThe purpose of Reverse Mathematics is to study the role of set ex-istence axioms,with an eye to determining which axioms are needed in order to prove speciﬁc mathematical theorems.In many cases,it is shown that a speciﬁc mathematical theorem is equivalent to the set existence axiom which is needed to prove it.Such equivalences are often proved in the weak base theory RCA0.RCA0may be viewed as a kind of formalized constructive or recursive mathematics,with full clas-sical logic but severely restricted comprehension and induction.The program of Reverse Mathematics has been developed in many publica-tions;see for instance[5,10,11,12,20].In this paper we carry out a Reverse Mathematics study of some aspects of classical Lebesgue measure theory.Historically,the subject of measure theory developed hand in hand with the nonconstructive, set-theoretic approach to mathematics.Errett Bishop has remarked that the foundations of measure theory present a special challenge to the constructive mathematician.Although our program of Reverse Mathematics is quite diﬀerent from Bishop-style constructivism,we feel that Bishop’s remark implicitly raises an interesting question:Which nonconstructive set existence axioms are needed for measure theory?VITALI’S THEOREM AND WWKL 2This paper,together with earlier papers of Yu and others [21,22,23,24,25,26],constitute an answer to that question.The results of this paper build upon and clarify some early results of Yu and Simpson.The reader of this paper will ﬁnd that familiarity with Yu–Simpson [26]is desirable but not essential.We begin in section 2by exploring the extent to which measure theory can be developed in RCA 0.We show that pairwise disjoint countable additivity for open sets of reals is provable in RCA 0.This is in contrast to a result of Yu–Simpson [26]:countable additivity for open sets of reals is equivalent over RCA 0to a nonconstructive set existence axiom known as Weak Weak K¨o nig’s Lemma (WWKL).We show in sections 3and 4that several other basic propositions of measure theory are also equivalent to WWKL over RCA 0.Finally in section 5we show that the Vitali Covering Theorem is likewise equivalent to WWKL over RCA 0.2.Measure Theory in RCA 0Recall that RCA 0is the subsystem of second order arithmetic with∆01comprehension and Σ01induction.The purpose of this section is toshow that some measure-theoretic results can be proved in RCA 0.Within RCA 0,let X be a compact separable metric space.We deﬁne C (X )= A,the completion of A ,where A is the vector space of rational “polynomials”over X under the sup-norm, f =sup x ∈X |f (x )|.For the precise deﬁnitions within RCA 0,see [26]and section III.E of Brown’s thesis [4].The construction of C (X )within RCA 0is inspired by the constructive Stone–Weierstrass theorem in section 4.5of Bishop and Bridges [2].It is provable in RCA 0that there is a natural one-to-one correspondence between points of C (X )and continuous functions f :X →R which are equipped with a modulus of uniform continuity ,that is to say,a function h :N →N such that for all n ∈N and x ,y ∈Xd (x,y )<12n .Within RCA 0we deﬁne a measure (more accurately,a nonnegative Borel probability measure)on X to be a nonnegative bounded linear functional µ:C (X )→R such that µ(1)=1.(Here µ(1)denotes µ(f ),f ∈C (X ),f (x )=1for all x ∈X .)For example,if X =[0,1],the unit interval,then there is an obvious measure µL :C ([0,1])→R given by µL (f )= 10f (x )dx ,the Riemann integral of f from 0to 1.We refer to µL as Lebesgue measure on [0,1].There is also the obvious generalization to Lebesgue measure µL on X =[0,1]n ,the n -cube.VITALI’S THEOREM AND WWKL 3Deﬁnition 2.1(measure of an open set).This deﬁnition is made in RCA 0.Let X be any compact separable metric space,and let µbe any measure on X .If U is an open set in X ,we deﬁneµ(U )=sup {µ(f )|f ∈C (X ),0≤f ≤1,f =0on X \U }.Within RCA 0this supremum need not exist as a real number.(Indeed,the existence of µ(U )for all open sets U is equivalent to ACA 0over RCA 0.)Therefore,when working within RCA 0,we interpret assertions about µ(U )in a “virtual”or comparative sense.For example,µ(U )≤µ(V )is taken to mean that for all >0and all f ∈C (X )with 0≤f ≤1and f =0on X \U ,there exists g ∈C (X )with 0≤g ≤1and g =0on X \V such that µ(f )≤µ(g )+ .See also [26].Some basic properties of Lebesgue measure are easily proved in RCA 0.For instance,it is straightforward to show that the Lebesgue measure of the union of a ﬁnite set of pairwise disjoint open intervals is equal to the sum of the lengths of the intervals.We deﬁne L 1(X,µ)to be the completion of C (X )under the L 1-norm given by f 1=µ(|f |).(For the precise deﬁnitions,see [5]and[26].)In RCA 0we see that L 1(X,µ)is a separable Banach space,but to assert within RCA 0that points of the Banach space L 1(X,µ)represent measurable functions f :X →R is problematic.We shall comment further on this question in section 4below.Lemma 2.2.The following is provable in RCA 0.If U n ,n ∈N ,is a sequence of open sets,then µ∞ n =0U n ≥lim k →∞µ k n =0U n .Proof.Trivial.Lemma 2.3.The following is provable in RCA 0.If U 0,U 1,...,U k is a ﬁnite,pairwise disjoint sequence of open sets,then µ k n =0U n ≥k n =0µ(U n ).Proof.Trivial.An open set is said to be connected if it is not the union of two disjoint nonempty open sets.Let us say that a compact separable metric space X is nice if for all suﬃciently small δ>0and all x ∈X ,the open ballB (x,δ)={y ∈X |d (x,y )<δ}VITALI’S THEOREM AND WWKL4 is connected.Such aδis called a modulus of niceness for X.For example,the unit interval[0,1]and the n-cube[0,1]n are nice, but the Cantor space2N is not nice.Theorem2.4(disjoint countable additivity).The following is prov-able in RCA0.Assume that X is nice.If U n,n∈N,is a pairwise disjoint sequence of open sets in X,thenµ∞n=0U n=∞n=0µ(U n).Proof.Put U= ∞n=0U n.Note that U is an open set.By Lemmas2.2and2.3,we have in RCA0thatµ(U)≥ ∞n=0µ(U n).It remainsto prove in RCA0thatµ(U)≤ ∞n=0µ(U n).Let f∈C(X)be suchthat0≤f≤1and f=0on X\U.It suﬃces to prove thatµ(f)≤∞n=0µ(U n).Claim1:There is a sequence of continuous functions f n:X→R, n∈N,deﬁned by f n(x)=f(x)for all x∈U n,f n(x)=0for all x∈X\U n.To prove this in RCA0,recall from[6]or[20]that a code for a continuous function g from X to Y is a collection G of quadruples (a,r,b,s)with certain properties,the idea being that d(a,x)<r im-plies d(b,g(x))≤s.Also,a code for an open set U is a collection of pairs(a,r)with certain properties,the idea being that d(a,x)<r im-plies x∈U.In this case we write(a,r)<U to mean that d(a,b)+r<s for some(b,s)belonging to the code of U.Now let F be a code for f:X→R.Deﬁne a sequence of codes F n,n∈N,by putting(a,r,b,s) into F n if and only if1.(a,r,b,s)belongs to F and(a,r)<U n,or2.(a,r,b,s)belongs to F and b−s≤0≤b+s,or3.b−s≤0≤b+s and(a,r)<U m for some m=n.It is straightforward to verify that F n is a code for f n as required by claim1.Claim2:The sequence f n,n∈N,is a sequence of elements of C(X). To prove this in RCA0,we must show that the sequence of f n’s has a sequence of moduli of uniform continuity.Let h:N→N be a modulus of uniform continuity for f,and let k be so large that1/2k is a modulus of niceness for X.We shall show that h :N→N deﬁned by h (m)=max(h(m),k)is a modulus of uniform continuity for all of the f n’s.Let x,y∈X and m∈N be such that d(x,y)<1/2h (m). To show that|f n(x)−f n(y)|<1/2m,we consider three cases.Case1:VITALI’S THEOREM AND WWKL5 x,y∈U n.In this case we have|f n(x)−f n(y)|=|f(x)−f(y)|<1VITALI’S THEOREM AND WWKL 6From (1)we see that for each >0there exists k such that µ(f )− ≤ kn =0µ(f n ).Thus we haveµ(f )− ≤kn =0µ(f n )≤k n =0µ(U n )≤∞ n =0µ(U n ).Since this holds for all >0,it follows that µ(f )≤ ∞n =0µ(U n ).Thus µ(U )≤ ∞n =0µ(U n )and the proof of Theorem 2.4is complete.Corollary 2.5.The following is provable in RCA 0.If (a n ,b n ),n ∈N is a sequence of pairwise disjoint open intervals,then µL ∞ n =0(a n ,b n ) =∞ n =0|a n −b n |.Proof.This is a special case of Theorem 2.4.Remark 2.6.Theorem 2.4fails if we drop the assumption that X is nice.Indeed,let µC be the familiar “fair coin”measure on the Cantor space X =2N ,given by µC ({x |x (n )=i })=1/2for all n ∈N and i ∈{0,1}.It can be shown that disjoint ﬁnite additivity for µC is equivalent to WWKL over RCA 0.(WWKL is deﬁned and discussed in the next section.)In particular,disjoint ﬁnite additivity for µC is not provable in RCA 0.3.Measure Theory in WWKL 0Yu and Simpson [26]introduced a subsystem of second order arith-metic known as WWKL 0,consisting of RCA 0plus the following axiom:if T is a subtree of 2<N with no inﬁnite path,thenlim n →∞|{σ∈T |length(σ)=n }|VITALI’S THEOREM AND WWKL 7see also Sieg [18].In this sense,every mathematical theorem provable in WKL 0or WWKL 0is ﬁnitistically reducible in the sense of Hilbert’s Program;see [19,6,20].Remark 3.2.The study of ω-models of WWKL 0is closely related to the theory of 1-random sequences,as initiated by Martin-L¨o f [16]and continued by Kuˇc era [7,13,14,15].At the time of writing of [26],Yu and Simpson were unaware of this work of Martin-L¨o f and Kuˇc era.The purpose of this section and the next is to review and extend the results of [26]and [21]concerning measure theory in WWKL 0.A measure µ:C (X )→R on a compact separable metric space X is said to be countably additive if µ∞ n =0U n =lim k →∞µ k n =0U n for any sequence of open sets U n ,n ∈N ,in X .The following theorem is implicit in [26]and [21].Theorem 3.3.The following assertions are pairwise equivalent over RCA 0.1.WWKL.2.(countable additivity)For any compact separable metric space Xand any measure µon X ,µis countably additive.3.For any covering of the closed unit interval [0,1]by a sequence of open intervals (a n ,b n ),n ∈N ,we have ∞n =0|a n −b n |≥1.Proof.That WWKL implies statement 2is proved in Theorem 1of [26].The implication 2→3is trivial.It remains to prove that statement 3implies WWKL.Reasoning in RCA 0,let T be a subtree of 2<N with no inﬁnite path.PutT ={σ i |σ∈T,σ i /∈T,i <2}.For σ∈2<N put lh(σ)=length of σanda σ=lh(σ)−1n =0σ(n )2lh(σ).Note that |a σ−b σ|=1/2lh(σ).Note also that σ,τ∈2<N are incompa-rable if and only if (a σ,b σ)∩(a τ,b τ)=∅.In particular,the intervals (a τ,b τ),τ∈ T,are pairwise disjoint and cover [0,1)except for some of the points a σ,σ∈2<N .Fix >0and put c σ=a σ− /4lh(σ),d σ=a σ+ /4lh(σ).Then the open intervals (a τ,b τ),τ∈ T,(c σ,d σ),VITALI’S THEOREM AND WWKL 8σ∈2<N and (1− ,1+ )form a covering of [0,1].Applying statement 3,we see that the sum of the lengths of these intervals is ≥1,i.e. τ∈ T12lh(τ)=1.From this,equation (2)follows easily.Thus we have proved that state-ment 3implies WWKL.This completes the proof of the theorem.It is possible to take a somewhat diﬀerent approach to measure the-ory in RCA 0.Note that the deﬁnition of µ(U )that we have given (Deﬁnition 2.1)is extensional in RCA 0.This means that if U and V contain the same points then µ(U )=µ(V ),provably in RCA 0.An alternative approach is the intensional one,embodied in Deﬁnition 3.4below.Recall that an open set U is given in RCA 0as a sequence of basic open sets.In the case of the real line,basic open sets are just intervals with rational endpoints.Deﬁnition 3.4(intensional Lebesgue measure).We make this deﬁni-tion in RCA 0.Let U = (a n ,b n ) n ∈N be an open set in the real line.The intensional Lebesgue measure of U is deﬁned by µI (U )=lim k →∞µL k n =0(a n ,b n ) .Theorem 3.5.It is provable in RCA 0that intensional Lebesgue mea-sure µI is countably additive on open sets.In other words,if U n ,n ∈N ,is a sequence of open sets,then µI∞ n =0U n =lim k →∞µI k n =0U n .Proof.This is immediate from the deﬁnitions,since ∞n =0U n is deﬁned as the union of the sequences of basic open intervals in U n ,n ∈N .Returning now to WWKL 0,we can prove that intensional Lebesgue measure concides with extensional Lebesgue measure.In fact,we have the following easy result.Theorem 3.6.The following assertions are pairwise equivalent over RCA 0.VITALI’S THEOREM AND WWKL91.WWKL.2.µI(U)=µL(U)for all open sets U⊆[0,1].3.µI is extensional on open sets.In other words,for all open setsU,V⊆[0,1],if∀x(x∈U↔x∈V)thenµI(U)=µI(V).4.For all open sets U⊇[0,1],we haveµI(U)≥1.Proof.This is immediate from Theorems3.3and3.5.4.More Measure Theory in WWKL0In this section we show that a good theory of measurable functions and measurable sets can be developed within WWKL0.Weﬁrst consider pointwise values of measurable functions.Our ap-proach is due to Yu[21,24].Let X be a compact separable metric space and letµ:C(X)→R be a positive Borel probability measure on X.Recall that L1(X,µ)is deﬁned within RCA0as the completion of C(X)under the L1-norm.In what sense or to what extent can we prove that a point of the Banach space L1(X,µ)gives rise to a function f:X→R?In order to answer this question,recall that f∈L1(X,µ)is given by a sequence f n∈C(X),n∈N,which converges to f in the L1-norm; more preciselyf n−f n+1 1≤12nfor all n,and|f m(x)−f m (x)|≤12k.VITALI’S THEOREM AND WWKL10 Then for x∈C fnand m ≥m≥n+2k+2we have|f m(x)−f m (x)|≤m −1i=m|f i(x)−f i+1(x)|≤∞i=n+2k+2|f i(x)−f i+1(x)|≤12k.We need a lemma:Lemma4.2.The following is provable in RCA0.For f∈C(X)and >0,we haveµ({x|f(x)> })≤ f 1/ .Proof.Put U={x|f(x)> }.Note that U is an open set.If g∈C(X),0≤g≤1,g=0on X\U,then we have g≤|f|, hence µ(g)=µ( g)≤µ(|f|)= f 1,henceµ(g)≤ f 1/ .Thus µ(U)≤ f 1/ and the lemma is proved.Using this lemma we haveµ(X\C fnk )=µx∞i=n+2k+2|f i(x)−f i+1(x)|>12i=1VITALI’S THEOREM AND WWKL 11hence by countable additivityµ(X \C f n )≤∞ k =0µ(X \C f nk )≤∞k =012n .This completes the proof of Proposition 4.1.Remark 4.3(Yu [21]).In accordance with Proposition 4.1,forf = f n n ∈N ∈L 1(X,µ)and x ∈ ∞n =0C f n ,we deﬁne f (x )=lim n →∞f n (x ).Thus we see thatf (x )is deﬁned on an F σset of measure 1.Moreover,if f =g in L 1(X,µ),i.e.if f −g 1=0,then f (x )=g (x )for all x in an F σset of measure 1.These facts are provable in WWKL 0.We now turn to a discussion of measurable sets within WWKL 0.We sketch two approaches to this topic.Our ﬁrst approach is to identify measurable sets with their characteristic functions in L 1(X,µ),accord-ing to the following deﬁnition.Deﬁnition 4.4.This deﬁnition is made within WWKL 0.We say that f ∈L 1(X,µ)is a measurable characteristic function if there exists a sequence of closed setsC 0⊆C 1⊆···⊆C n ⊆...,n ∈N ,such that µ(X \C n )≤1/2n for all n ,and f (x )∈{0,1}for all x ∈ ∞n =0C n .Here f (x )is as deﬁned in Remark 4.3.Our second approach is more direct,but in its present form it applies only to certain speciﬁc situations.For concreteness we consider only Lebesgue measure µL on the unit interval [0,1].Our discussion can easily be extended to Lebesgue measure on the n -cube [0,1]n ,the “fair coin”measure on the Cantor space 2N ,etc .Deﬁnition 4.5.The following deﬁnition is made within RCA 0.Let S be the Boolean algebra of ﬁnite unions of intervals in [0,1]with rational endpoints.For E 1,E 2∈S we deﬁne the distanced (E 1,E 2)=µL ((E 1\E 2)∪(E 2\E 1)),the Lebesgue measure of the symmetric diﬀerence of E 1and E 2.Thus d is a pseudometric on S ,and we deﬁne S to be the compact separable metric space which is the completion of S under d .A point E ∈ S is called a Lebesgue measurable set in [0,1].VITALI’S THEOREM AND WWKL 12We shall show that these two approaches to measurable sets (Deﬁ-nitions 4.4and 4.5)are equivalent in WWKL 0.Begin by deﬁning an isometry χ:S →L 1([0,1],µL )as follows.For 0≤a <b ≤1deﬁneχ([a,b ])= f n n ∈N ∈L 1([0,1],µL )where f n (0)=f n (a )=f n (b )=f n (1)=0and f n a +b −a 2n +1=1and f n ∈C ([0,1])is piecewise linear otherwise.Thus χ([a,b ])is a measurable characteristic function corresponding to the interval [a,b ].For 0≤a 1<b 1<···<a k <b k ≤1deﬁneχ([a 1,b 1]∪···∪[a k ,b k ])=χ([a 1,b 1])+···+χ([a k ,b k ]).It is straightforward to prove in RCA 0that χextends to an isometryχ: S→L 1([0,1],µL ).Proposition 4.6.The following is provable in WWKL 0.If E ∈ Sis a Lebesgue measurable set,then χ(E )is a measurable characteristic function in L 1([0,1],µL ).Conversely,given a measurable characteristic function f ∈L 1([0,1],µL ),we can ﬁnd E ∈ Ssuch that χ(E )=f in L 1([0,1],µL ).Proof.It is straightforward to prove in RCA 0that for all E ∈ S , χ(E )is a measurable characteristic function.For the converse,let f be a measurable characteristic function.By Deﬁnition 4.4we have that f (x )∈{0,1}for all x ∈ ∞n =0C n .ByProposition 4.1we have |f (x )−f 3n +3(x )|<1/2n for all x ∈C f n .Put U n ={x ||f 3n +3(x )−1|<1/2n }and V n ={x ||f 3n +3(x )|<1/2n }.Then for n ≥1,U n and V n are disjoint open sets.Moreover C n ∩C f n ⊆U n ∪V n ,hence µL (U n ∪V n )≥1−1/2n −1.By countable additivity(Theorem 3.3)we can eﬀectively ﬁnd E n ,F n ∈S such that E n ⊆U n and F n ⊆V n and µL (E n ∪F n )≥1−1/2n −2.Put E = E n +5 n ∈N .It is straightforward to show that E belongs to S and that χ(E )=f in L 1([0,1],µL ).This completes the proof.Remark 4.7.We have presented two notions of Lebesgue measurable set and shown that they are equivalent in WWKL 0.Our ﬁrst notion (Deﬁnition 4.4)has the advantage of generality in that it applies to any measure on a compact separable metric space.Our second no-tion (Deﬁnition 4.5)is advantageous in other ways,namely it is more straightforward and works well in RCA 0.It would be desirable to ﬁnd a single deﬁnition which combines all of these advantages.VITALI’S THEOREM AND WWKL 135.Vitali’s TheoremLet S be a collection of sets.A point x is said to be Vitali covered by S if for all >0there exists S ∈S such that x ∈S and the diameter of S is less than .The Vitali Covering Theorem in its simplest form says the following:if I is a sequence of intervals which Vitali covers an interval E in the real line,then I contains a countable,pairwise disjoint set of intervals I n ,n ∈N ,such that ∞n =0I n covers E except for a set of Lebesgue measure 0.The purpose of this section is to show that various forms of the Vitali Covering Theorem are provable in WWKL 0and in fact equivalent to WWKL over RCA 0.Throughout this section,we use µto denote Lebesgue measure.Lemma 5.1(Baby Vitali Lemma).The following is provable in RCA 0.Let I 0,...,I n be a ﬁnite sequence of intervals.Then we can ﬁnd a pair-wise disjoint subsequence I k 0,...,I k m such thatµ(I k 0∪···∪I k m )≥1VITALI’S THEOREM AND WWKL 14I =[2a −b,2b −a ].)Thusµ(I 0∪···∪I n )≤µ(I k 0∪···∪I k m )≤µ(I k 0)+···+µ(I k m )=3µ(I k 0)+···+3µ(I k m )=3µ(I k 0∪···∪I k m )and the lemma is proved.Lemma 5.2.The following is provable in WWKL 0.Let E be an in-terval,and let I n ,n ∈N ,be a sequence of intervals.If E ⊆ ∞n =0I n ,then µ(E )≤lim k →∞µ k n =0I n .Proof.If the intervals I n are open,then the desired conclusion follows immediately from countable additivity (Theorem 3.3).Otherwise,ﬁx >0and let I n be an open interval with the same midpoint as I n andµ(I n )=µ(I n )+µ(E \A ).(3)VITALI’S THEOREM AND WWKL 15To prove the claim,use Lemma 5.2and the Vitali property to ﬁnd a ﬁnite set of intervals J 1,...,J l ∈I such that J 1,...,J l ⊆E \A andµ(E \(A ∪J 1∪···∪J l ))<13µ(J 1∪···∪J l ).We then have µ(E \(A ∪I 1∪···∪I k ))<212µ(E \A )≤212µ(E \A )=34nµ(E ).Then by countable additivity we have µ E \∞ n =1A n =0and the lemma is proved.Remark 5.4.It is straightforward to generalize the previous lemma to the case of a Vitali covering of the n -cube [0,1]n by closed balls or n -dimensional cubes.In the case of closed balls,the constant 3in the Baby Vitali Lemma 5.1is replaced by 3n .Theorem 5.5.The Vitali theorem for the interval [0,1](as stated in Lemma 5.3)is equivalent to WWKL over RCA 0.Proof.Lemma 5.3shows that,in RCA 0,WWKL implies the Vitali theorem for intervals.It remains to prove within RCA 0that the Vitali theorem for [0,1]implies WWKL.Instead of proving WWKL,we shall prove the equivalent statement 3.3.3.Reasoning in RCA 0,suppose thatVITALI’S THEOREM AND WWKL 16(a n ,b n ),n ∈N ,is a sequence of open intervals which covers [0,1].Let I be the countable set of intervals (a nki ,b nki )= a n +i k(b n −a n ) where i,k,n ∈N and 0≤i <k .Then I is a Vitali covering of [0,1].By the Vitali theorem for intervals,I contains a sequence of pairwise disjoint intervals I m ,m ∈N ,such that µ ∞ m =0I m ≥1.By disjoint countable additivity (Corollary 2.5),we have∞m =0µ(I m )≥1.From this it follows easily that∞n =0|a n −b n |≥1.Thus we have 3.3.3and our theorem is proved.We now turn to Vitali’s theorem for measurable sets.Recall our discussion of measurable sets in section 4.A sequence of intervals I is said to almost Vitali cover a Lebesgue measurable set E ⊆[0,1]if for all >0we have µL (E \O )=0,where O = {I |I ∈I ,diam(I )< }.Theorem 5.6.The following is provable in WWKL 0.Let E ⊆[0,1]be a Lebesgue measurable set with µ(E )>0.Let I be a sequence of intervals which almost Vitali covers E .Then I contains a pairwise disjoint sequence of intervals I n ,n ∈N ,such that µ E \∞ n =0I n =0.Proof.The proof of this theorem is similar to that of Lemma 5.3.The only modiﬁcation needed is in the proof of the claim.Recall from Deﬁnition 4.5that E =lim n →∞E n where each E n is a ﬁnite union of intervals in [0,1].Fix m so large thatµ((E \E m )∪(E m \E ))<1VITALI’S THEOREM AND WWKL 17andµ(E m \(A ∪J 1∪···∪J l ))<136µ(E \A )<236µ(E \A )≤236µ(E \A )<236µ(E \A )=3,The Baire category theorem in weak subsystems of second order arith-metic ,Journal of Symbolic Logic 58(1993),557–578.7.O.Demuth and A.Kuˇc era,Remarks on constructive mathematical analysis ,[3],1979,pp.81–129.8.H.-D.Ebbinghaus,G.H.M¨u ller,and G.E.Sacks (eds.),Recursion Theory Week ,Lecture Notes in Mathematics,no.1141,Springer-Verlag,1985,IX +418pages.VITALI’S THEOREM AND WWKL189.Harvey Friedman,unpublished communication to Leo Harrington,1977.10.Harvey Friedman,Stephen G.Simpson,and Rick L.Smith,Countable algebraand set existence axioms,Annals of Pure and Applied Logic25(1983),141–181.11.,Randomness and generalizations ofﬁxed point free functions,[1],1990, pp.245–254.15.,Subsystems of Second Order Arithmetic,Perspectives in Mathematical Logic,Springer-Verlag,1998,XIV+445pages.21.Xiaokang Yu,Measure Theory in Weak Subsystems of Second Order Arithmetic,Ph.D.thesis,Pennsylvania State University,1987,vii+73pages.22.,Riesz representation theorem,Borel measures,and subsystems of sec-ond order arithmetic,Annals of Pure and Applied Logic59(1993),65–78. 24.,A study of singular points and supports of measures in reverse mathe-matics,Annals of Pure and Applied Logic79(1996),211–219.26.Xiaokang Yu and Stephen G.Simpson,Measure theory and weak K¨o nig’slemma,Archive for Mathematical Logic30(1990),171–180.E-mail address:dkb5@,giusto@dm.unito.it,simpson@ The Pennsylvania State University。

Abstract

1
Int roduct
ion
This paper studies the problem of constructing simplicial complexes that represent or approximate a geometric object in some finite-dimensional Euclidean space, Rd. We refer to the geometric object as a topological space or subspace of Rd. This problem arises in geometric modeling and finite element analysis, and it is a special case of the grid generation problem [21]. It is special because we only consider grids or complexes made up of simplices. The problem can be divided into two questions: How do we choose the points or vertices of the grid? How do we connect the vertices using edges, triangles, and higher-dimensional simplices?
in a non-empty set. By the nerve theorem, ~Dx and X are homotopy equivalent if all such sets are contractible.

Markus表征理论对英汉心理词典表征模式的解释力

Markus 表征理论对英汉心理词典表征模式的解释力陈士法,彭建武(山东科技大学外国语学院,山东青岛266510)摘要:我们使用英语中的单词素词和多词素词作为不同类型的单词来设计实验,验证了M arkus 双语心理词典表征理论对英汉双语心理词典表征模式的解释力。

实验证明,在初级水平英汉双语者的大脑中,不可能只存在词汇联结模式;同样,在高级水平英汉双语者的大脑中,也不可能只存在语义中介模式。

影响双语心理词典表征的因素较多,若仅从一个或几个方面来研究双语心理词典的表征模式,就难免挂一而漏万。

关键词:英汉双语;心理词典;词汇联结模式;语义中介模式中图分类号:H 0 05 文献标示码:A 文章编号:1008 7699(2009)02 0074 06The Explanatory Power of Markus s Representation Theory toEnglish Chinese Bilingual Mental LexiconCH EN Shi fa,PENG Jian wu(Colle ge of F ore ig n L anguag es ,S hand ong Unive rsity of S cience and T echnolog y ,Qingd ao,S hand ong 266510,China)Abstract:T he Eng lish mono mo rphemic w or ds and poly mo rphemic w or ds are employ ed as materials in a lex ical experiment to verify the ex planato ry pow er of M arkus representation theor y about bilingual ment al lex icon to EnglishChinese biling ual menta l lex icon.T he r esults confir m us that there sho uld not be o nly an L1 wo rd for m linking model in elementar y bilinguals mental lex ico n.A nd there should no t be o nly a concept mediat ing mo del in adv anced bilinguals mental lex icon.I t is likely to miss the point if o nly one or sever al aspects of biling ual mental lex icon ar etaken into co nsideration.Key words:Eng lish Chinese biling uals;mental lexicon;an L 1 wo rd for m linking mo del;a concept mediating model收稿日期:2009 01 12基金项目:山东省人文社会科学研究项目(S07YF21)作者简介:陈士法(1967 ),男,山东济宁人,山东科技大学外国语学院教授. Garnham 认为,心理词典是词汇知识的心理状态,它标注了一个单词的拼写、发音、词类和意义等信息。

QP-problems

1
where f 2 C2(IRn) is a convex function. For numerical testing we use the easier
quadratic form (1). Since f is convex, the Karush-Kuhn-Tucker conditions are necessary and su cient for x being a global minimizer of (2). Friedlander et.al. use the KKT-conditions to construct an exact penalty function which de nes a "primal-dual" box-constrained optimization problem with 2n + m variables. In the third section their Trust-Region-SQP method for solving the penalty problem is described. For testing this combination of penalty-function and Trust-Region-SQP approach a special problem generator is used. Numerical results are presented.
and an initial trust region radius 0 min.
Step 1:(upper bound of Bk)
Set the trust-region-radius :=
Step 2:(compute initial step)

并联机床工件自动定位

收稿日期:20010606基金项目:博士学科点专项科研基金(98021332)和黑龙江省科学技术委员会重大基金资助作者简介:卓桂荣(1968-),女(汉),山东,工程师,博士研究生卓桂荣文章编号:100328728(2002)0420600202并联机床工件自动定位卓桂荣,王知行,刘文涛(哈尔滨工业大学机电学院,哈尔滨　150001)摘　要:根据并联机床的几何约束条件,提出并联机床工件自动定位的算法。

该算法简单实用,对一个复杂曲面可以自动找到其在并联机床工作空间中可以加工的位置。

关　键　词:并联机床;自动定位;Stew art 平台中图分类号:T G 54;T P 242.2 文献标识码:AOn Self -Or ien ta tion of W orkp ieces i n a Para llelM ach i ne ToolZHUO Gu i 2rong ,W AN G Zh i 2x ing ,L I U W en 2tao(Schoo l of M echan ical and E lectrical Engineering ,H arb in In stitu te of T echno logy ,H arb in 150001)Abstract :A m ethod is p resen ted fo r self 2o rien tati on of w o rkp ieces in a parallel m ach ine too l acco rding to the geo 2m etrical restricti on of the parallel m ach ine too l .T he m ethod is si m p le and u sefu l ,in w h ich a comp lex concave 2con 2vex su rface can find its m ach in ing locati on at the w o rk ing space in the parallel m ach ine too l au tom atically .Key words :Parallel m ach ine too l ;Self 2o rien tati on ;Stew art p latfo rm 1　工件自动定位原理并联机床工作空间相对较小且计算复杂,在加工像叶轮、叶片等具有复杂曲面的工件时,如何设置工件的加工位置和空间姿态,以保证工件在工作空间内,使机床刀具能够达到加工面上的所有可能加工点,这就是工件在并联机床工作空间中的定位问题。

数字化时代的知识传播与版权保护_李春茜

数字化时代的知识传播与版权保护李春茜(中山大学信息管理系,广州510275)摘　要　本文从知识传播、版权保护以及二者之间的关系研究入手,在此基础上深入地探讨了数字化时代的知识传播与版权保护的特点与关系,以及版权保护所面临的新的问题和挑战。

关键词　数字化时代　著作权　知识传播Knowledge Spread and Copyr ight Protection i n D ig ita l EraL i Chunqian(D epartm ent of Info r m ati on M anagem ent ,Zhongshan U niversity ,Guangzhou 510275)Abstract T he paper begins w ith the relati ons betw een the know ledge sp read and copyrigh t p ro tecti on ,onth is basis it discusses deep ly the characteristics of the know ledge sp read and copyrigh t p ro tecti on ,their rela 2ti ons ,and the new p roblem s and challenges that the copyrigh t p ro tecti on w ill face .Keywords D igital era Copyrigh t law Know ledge sp read收稿日期:2000-6-25 在以印刷型文献为主要知识传播手段的时代,版权法(在我国称为“著作权法”)使版权垄断与信息共亨保持了一定的平衡,从而在较为有效地保护知识产权的同时,不会影响知识的传播和利用。

但是进入数字化时代,先进的信息技术正在打破已有的平衡,网络上的电子化信息的传播给版权保护带来了新的挑战。

基于曲轴瞬时加速度分析的发动机启动过程着火判定与应用

基于曲轴瞬时加速度分析的发动机启动过程着火判定与应用3杨福源张京永王晓光周明欧阳明高( 清华大学 ,汽车安全与节能国家重点实验室 ,北京 100084)[ 摘要 ] 提出不同于失火判定的基于曲轴瞬时加速度分析的发动机启动过程着火判定方法 ,即通过两次启动过程的瞬时加速度对比寻找喷油启动过程的着火始点。

理论分析和试验表明该方法不仅可以准确识别着火始点 ,而且识别精度也从循环级提高到角度级。

试验在一台 6 缸电控共轨柴油机上进行。

叙词 :电控发动机 ,启动过程 ,燃烧始点判定 ,曲轴瞬时加速度J udgeme nt and Applicatio n of Ignitio n Mo m e nt Duri n g Engi ne St art upPerio d Based o n Crankshaf t Transie nt Acceleratio n Anal ysisYang Fuyuan , Z hang Jingyong , Wang Xiaoguang , Z hou Ming & Ouyang MinggaoTsi n g h u a U ni versi t y , S t ate Key L aborat ory of A ut om o t i ve S af et y a n d Energy , Bei j i n g 100084Abstract This paper brings fo rward an igniti o n detecti o n met h o d based o n crankshaf t t r ansient accelera 2ti o n analysis 1 It co m pares t h e t r ansient accelerati o n of t w o start u p p r ocedures and t h en find o u t t h e igniti o n m o 2ment ( start of co mbusti o n ) during start up peri o d 1Crankshaf t dynamics analysis and real test show t hat t h is met ho d can identi f y t he igniti o n m o ment ,wit h t he accuracy imp roved f ro m cycle level to angle level 1 The test is carried o u t o n a 6 - cylinder co m m o n rail diesel engine 1K ey w ords :E lectron ically controlled engine , S tartup period , I gn i t i on mom ent , C ran k shaf t transient acceler 2at i on将可对柴油机启动过程的喷油量、喷油正时、喷射压力等参数进行优化。

New Characterizations of Input to State Stability

New Characterizations of Input to State StabilityEduardo D.SontagYuan WangAbstract—We present new characterizations of the Input to State Stability property.As a consequence of these re-sults,we show the equivalence between the ISS property and several(apparent)variations proposed in the literature.I.IntroductionThis paper studies stability questions for systems of the general formΣ:˙x=f(x,u),(1) with states x(t)evolving in Euclidean space R n and con-trols u(·)taking values u(t)∈U⊆R m,for some positive integers n and m(in all the main results,U=R m).The questions to be addressed all concern the study of the size of each solution x(t)—its asymptotic behavior as well as maximum value—as a function of the initial condition x(0)and the magnitude of the control u(·).One of the most important issues in the study of control systems is that of understanding the dependence of state trajectories on the magnitude of inputs.This is especially relevant when the inputs in question represent disturbances acting on a system.For linear systems,this leads to the consideration of gains and the operator-theoretic approach, including the formulation of H∞control.For not necessar-ily linear systems,there is no complete agreement as yet regarding what are the most useful formulations of system stability with respect to input perturbations.One candi-date for such a formulation is the property called“input to state stability”(ISS),introduced in[12].Various authors, (see e.g.[4],[5],[6],[10],[17]have subsequently employed this property in studies ranging from robust control and highly nonlinear small-gain theorems to the design of ob-servers and the study of parameterization issues;for expo-sitions see[14]and most especially the textbooks[7],[8]. The ISS property is deﬁned in terms of a decay estimate of solutions,and is known(cf.[15])to be equivalent to the validity of a dissipation inequalitydV(x(t))dt≤σ(|u(t)|)−α(|x(t)|)holding along all possible trajectories(this is reviewed be-low),for an appropriate“energy storage”function V and comparison functionsσ,α.(A dual notion of“output-to-state stability”(OSS)can also be introduced,and leads to the study of nonlinear detectability;see[16].)E.Sontag is with SYCON-Rutgers Center for Systems and Control,Department of Mathematics,Rutgers University,New Brunswick,NJ08903.e-mail:sontag@.This re-search was supported in part by US Air Force Grant F49620-95-1-0101.Y.Wang is with Department of Mathematics,Florida Atlantic Uni-versity,Boca Raton,FL33431.e-mail:ywang@. This research was supported in part by NSF Grants DMS-9457826 and DMS-9403924In some cases,notably in[2],[6],[18],authors have suggested apparent variations of the ISS property,which are more natural when solving particular control problems. The main objective of this paper is to point out that such variations are in fact theoretically equivalent to the orig-inal ISS deﬁnition.(This does not in any way diminish the interest of these other authors’contributions;on the contrary,the alternative characterizations are of great in-terest,especially since the actual estimates obtained may be more useful in one form than another.For instance, the“small-gain theorems”given in[6],[2]depend,in their applicability,on having the ISS property expressed in a particular form.This paper merely states that from a the-oretical point of view,the properties are equivalent.For an analogy,the notion of“convergence”in R n is independent of the particular norm used—e.g.all L p norms are equiv-alent—but many problems are more naturally expressed in one norm than another.)One of the main conclusions of this paper is that the ISS property is equivalent to the conjunction of the following two properties:(i)asymptotic stability of the equilibrium x=0of the unforced system(that is,of the system deﬁned by Equation(1)with u≡0)and(ii)every trajectory of(1) asymptotically approaches a ball around the origin whose radius is a function of the supremum norm of the control being applied.We prove this characterization along with many others.Since it is not harder to do so,the results are proved in slightly more generality,for notions relative to an arbitrary compact attractor rather than the equilibrium x=0.A.Basic Deﬁnitions and NotationsEuclidean norm in R n or R m is denoted simply as|·|. More generally,we will study notions relative to nonempty subsets A of R n;for such a set A,|ξ|A=d(ξ,A)= inf{d(η,ξ),η∈A}denotes the point-to-set distance from ξ∈R n to A.(So for the special case A={0},|ξ|{0}=|ξ|.) We also let,for eachε>0and each set A:B(A,ε):={ξ||ξ|A<ε},B(A,ε):={ξ||ξ|A≤ε}. Most of the results to be given are new even for A={0}, so the reader may wish to assume this,and interpret|ξ|A simply as the norm ofξ.(We prefer to deal with arbi-trary A because of potential applications to systems with parameters as well as the“practical stability”results given in Section VI.)The map f:R n×R m→R n in(1)is assumed to be locally Lipschitz continuous.By a control or input we mean a measurable and locally essentially bounded function u: I→R m,where I is a subinterval of R which contains theorigin,so that u (t )∈U for almost all t .Given a system with control-value set U ,we often consider the same system but with controls restricted to take values in some subset O ⊆U ;we use M O for the set of all such controls.Given any control u deﬁned on an interval I and any ξ∈R n ,there is a unique maximal solution of the initial value problem ˙x =f (x,u ),x (0)=ξ.This solution is deﬁned on some maximal open subinterval of I ,and it is denoted by x (·,ξ,u ).(For convenience,we allow negative times t in the expression x (t,ξ,u ),even though the interest is in behavior for t ≥0.)A forward complete system is one such that,for each u deﬁned on I =R ≥0,and each ξ,the solution x (t,ξ,u )is deﬁned on the entire interval R ≥0.The L m ∞-norm (possibly inﬁnite)of a control u is denoted by u ∞.That is, u ∞is the smallest number c such that |u (t )|≤c for almost all t ∈I .Whenever the domain I of a control u is not speciﬁed,it will be understood that I =R ≥0.A function F :S →R deﬁned on a subset S of R n containing 0is positive deﬁnite if F (x )>0for all x ∈S ,x =0,and F (0)=0.It is proper if the preimage F −1(−D,D )is bounded,for each D >0.A function γ:R ≥0→R ≥0is of class N (or an “N function”)if it is continuous and nondecreasing;it is of class N 0(or an “N 0function”)if in addition it satisﬁes γ(0)=0.A function γ:R ≥0→R ≥0is of class K (or a “K function”)if it is continuous,positive deﬁnite,and strictly increasing,and is of class K ∞if it is also unbounded (equivalently,it is proper,or γ(s )→+∞as s →+∞).Finally,recall that β:R ≥0×R ≥0→R ≥0is said to be a function of class KL if for each ﬁxed t ≥0,β(·,t )is of class K and for each ﬁxed s ≥0,β(s,t )decreases to zero as t →∞.(The notations K ,K ∞,and KL are fairly standard;the notations N and N 0are introduced here for convenience.)B.A Catalog of PropertiesWe catalog several properties of control systems which will be compared in this paper.Much of the terminology —except for “ISS”and the names for properties of unforced systems —is not standard,and should be considered ten-tative.A zero-invariant set A for a system Σas in Equation (1)is a subset A ⊆R n with the property that x (t,ξ,0)∈A for all t ≥0and all ξ∈A ,where 0denotes the control which is identically equal to zero on R ≥0.From now on,all deﬁnitions are with respect to a given forward-complete system Σas in Equation (1),and a given compact zero-invariant set A for this system.The main deﬁnitions follow.We ﬁrst recall the deﬁnition of the (ISS)property:∃γ∈K ,β∈KL st :∀ξ∈R n ∀u (·)∀t ≥0|x (t,ξ,u )|A ≤β(|ξ|A ,t )+γ( u ∞).(ISS)This was the form of the original deﬁnition of (ISS)given in [12].It is known that a system is (ISS)if and only if it satisﬁes a dissipation inequality,that is to say,there exists a smooth V :R n →R ≥0and there are functions αi ∈K ∞,i =1,2,3and σ∈K so thatα1(|ξ|A )≤V (ξ)≤α2(|ξ|A )(2)and∇V (ξ)f (ξ,v )≤σ(|v |)−α3(|ξ|A )(3)for each ξ∈R n and v ∈R m .See [15],[14]for proofs and an exposition,respectively.A very useful modiﬁcation of this characterization due to [11]is the fact that the (ISS)property is also equivalent to the existence of a smooth V satisfying (2)and Equation (3)replaced by an estimate of the type ∇V (ξ)f (ξ,v )≤−V (ξ)−α3(|ξ|A ).(This can be understood as:“for some positive deﬁnite and proper functions y =V (x )and v =W (u )of states and outputs respectively,along all trajectories of the system we have ˙y =−y +v ”.)The main purpose of this paper is to establish further equivalences for the (ISS)property.It will be technically convenient to ﬁrst introduce a local version of the property (ISS),by requiring only that the estimate hold if the initial state and the controls are small,as follows:∃ρ>0,γ∈K ,β∈KL st :∀|ξ|A ≤ρ,∀ u ∞≤ρ|x (t,ξ,u )|A ≤β(|ξ|A ,t )+γ( u ∞)∀t ≥0.(LISS)Several standard properties of the “unforced”system ob-tained when u ≡0will appear as technical conditions.We review these now.The 0-global attraction property with re-spect to A (0-GATT)holds if every trajectory x (·)of the zero-input system(Σ0):˙x =f (x,0)(4)satisﬁes lim t →∞|x (t,ξ,0)|A →0;if this is merely requiredof trajectories with initial conditions satisfying |x (0)|A <ρ,for some ρ>0,we have the 0-local attraction property with respect to A (0-LATT).The 0-local stability property with respect to A (0-LS)means that for each ε>0there is a δ>0so that |ξ|A <δimplies that |x (t,ξ,0)|A <εfor all t ≥0.Finally,the 0-asymptotic stability property with respect to A (0-AS)is the conjunction of (0-LATT)and (0-LS),and the 0-global asymptotic stability property with respect to A (0-GAS)is the conjunction of (0-GATT)and (0-LS).Note that (0-GAS)is equivalent to the conjunction of (0-AS)and (0-GATT).It is useful (see e.g.[3],[12],[7])to express these properties in terms of comparison functions:∃β∈KL st :∀ξ∈R n ∀t ≥0|x (t,ξ,0)|A ≤β(|ξ|A ,t ).(0-GAS)and∃ρ>0,β∈KL st :∀|ξ|A <ρ∀t ≥0|x (t,ξ,0)|A ≤β(|ξ|A ,t )(0-AS)respectively.Next we introduce several new concepts.The limit prop-erty with respect to A holds if every trajectory must at some time get to within a distance of A which is a function of the magnitude of the input:∃γ∈N0st:∀ξ∈R n∀u(·)inft≥0|x(t,ξ,u)|A≤γ( u ∞).(LIM) Observe that,if this property holds,then it also holds with someγ∈K∞.However,the caseγ≡0will be of interest, since it corresponds to a notion of attraction for systems in which controls u are viewed as disturbances.The asymptotic gain property with respect to A holds if every trajectory must ultimately stay not far from A, depending on the magnitude of the input:∃γ∈N0st:∀ξ∈R n∀u(·)limt→+∞|x(t,ξ,u)|A≤γ( u ∞).(AG) Again,if the property holds,then it also holds with some γ∈K∞,but the caseγ≡0will be of interest later.The uniform asymptotic gain property with respect to A holds if the limsup in(AG)is attained uniformly with respect to initial states in compacts and all u:∃γ∈N0∀ε>0∀κ>0∃T=T(ε,κ)≥0st:∀|ξ|A≤κsupt≥T|x(t,ξ,u)|A≤γ( u ∞)+ε∀u(·).(UAG)The boundedness property with respect to A holds if bounded initial states and controls produce uniformly bounded trajectories:∃σ1,σ2∈N st:∀ξ∈R n∀u(·)supt≥0|x(t,ξ,u)|A≤max{σ1(|ξ|A),σ2( u ∞)}.(BND)(This is sometimes called the“UBIBS”or“uniform bounded-input bounded-state”property.)The global sta-bility property with respect to A holds if in addition small initial states and controls produce uniformly small trajec-tories:∃σ1,σ2∈N0st:∀ξ∈R n∀u(·)supt≥0|x(t,ξ,u)|A≤max{σ1(|ξ|A),σ2( u ∞)}.(GS)Observe that,if this property holds,then it also holds with bothσi∈K∞.The local stability property with respect to A holds if we merely require a local estimate of this type:∃δ>0,α1,α2∈N0st:∀|ξ|A≤δ∀ u ∞≤δsupt≥0|x(t,ξ,u)|A≤max{α1(|ξ|A),α2( u ∞)}.(LS)If this property holds,then it also holds with bothαi∈K∞,i=1,2Theorem1:Assume given any forward-complete system Σas in Equation(1),with U=R m,and a compact zero-invariant set A for this system.The following properties are equivalent:A.(ISS)B.(LIM)&(0-AS)C.(UAG)D.(LIM)&(0-GAS)E.(AG)&(0-GAS)F.(AG)&(LISS)G.(AG)&(LS)H.(LIM)&(LS)I.(LIM)&(GS)J.(AG)&(GS)This theorem will follow from a several technical facts which are stated in the next section and proved later in the paper.These technical results are of interest in themselves.C.List of Main Technical StepsWe assume given a forward-complete systemΣas in Equation(1),with U=R m,and a compact zero-invariant set A for this system.For ease of reference,weﬁrst list several obvious implications:(UAG)=⇒(AG).(5)(AG)=⇒(LIM).(6)(ISS)=⇒(0-GAS).(7)(LISS)=⇒(0-AS).(8)(LISS)=⇒(LS).(9) Because(LIM)implies(0-GATT)and(0-GAS)is the same as(0-AS)plus(0-GATT),we have:(LIM)&(0-GAS)⇐⇒(LIM)&(0-AS).(10)It was shown in[15]that(ISS)⇐⇒(UAG)&(LS).(11)It turns out that(LS)is redundant,so(UAG)is in fact equivalent to(ISS):Proposition I.1:(UAG)⇒(LS).This observation generalizes a result which is well-known for systems with no controls(for which see e.g.[1,Theo-rem1.5.28]or[3,Theorem38.1]).It should be noted that the standing hypothesis that A is compact is essential for this implication;in the general case of noncompact sets A, the local stability property with respect to A is not redun-dant.From Proposition I.1and Equation(7),we know then that:(UAG)=⇒(0-GAS).(12) We also prove these results:Lemma I.2:(0-GAS)=⇒(LISS).Lemma I.3:(BND)&(LS)⇐⇒(GS).Lemma I.4:(LIM)&(GS)⇐⇒(AG)&(GS). Lemma I.5:(LIM)⇒(BND).The converse of Lemma I.5is of course false,as illustrated by the autonomous system˙x=0(with n=m=1),which even satisﬁes(GS)but does not satisfy(LIM).From Lem-mas I.3and I.5,we have that:(LIM)&(LS)⇐⇒(LIM)&(GS).(13)The most interesting technical result will be this: Proposition I.6:(LIM)&(LS)⇒(UAG).We now indicate how the proof of Theorem1follows from all these technical facts.•(A⇐⇒C):by Proposition I.1and Equation(11).•(C⇒E):by(5)and(12).•(E⇒F):by Lemma I.2.•(F⇒G):by Equation(9).•(G⇒H):by Equation(6).•(H⇒I):by Equation(13).•(I⇒J):by Lemma I.4.•(J⇒G):obvious.•(H⇒C):this is Proposition I.6.•(E⇒D):by Equation(6).•(B⇐⇒D):by Equation(10).•(D⇒H):by Lemma I.2and Equation(9).A very particular consequence of the main Theorem is worth focusing upon:A⇐⇒J,i.e.(ISS)is equivalent to having both the global stability property with respect to A and the asymptotic gain property with respect to A. Consider this property:∃γ∈N0st:∀ξ∈R n∀u(·)lim t→+∞|x(t,ξ,u)|A≤γlimt→+∞|u(t)|(14)(the limsup being understood in the“essential”sense,of holding up to a set of measure zero;note also that sinceγis continuous and nondecreasing,the right-hand term equalslim t→+∞γ(|u(t)|)).It is easy to show(see Lemma(II.1))thatthis is equivalent to(AG).The conjunction of(14)and(GS)is the“asymptotic L∞stability property”proposedby Teel and discussed in the survey paper[2](in that paper, A={0});it thus follows that asymptotic L∞stability is precisely the same as(ISS).In[18],Tsinias considered the following property(in thatpaper,A={0}):∃γ∈K st:∀ξ∈R n∀u(·)[|x(t,ξ,u)|A≥γ(|u(t)|)∀t≥0]⇒limt→∞|x(t,ξ,u)|A=0(15)which obviously implies(LIM).The author considered the conjunction of(15)and(LS)(more precisely,the author also assumed a local stability property that implies(LS), namely f(x,u)=Ax+Bu+o(x,u),with A Hurwitz); because of the equivalence A⇐⇒H,this conjunction is also equivalent to(ISS).The outline of the rest of the paper is as follows.In Section II weﬁrst prove Proposition I.1,Lemmas I.2, I.3,and I.4,and the equivalence between Property(14) and(AG),all of which are elementary.Section III con-tains the proof of the basic technical step needed to prove the main result,as well as a proof of Lemma I.5.After this,Section IV establishes a result showing that uniform global asymptotic stability of systems with disturbances(or equivalently,of an associated diﬀerential inclusion)follows from the non-uniform variant of the concept;this would appear to be a rather interesting result in itself,and in any case it is used in Section V to provide the proof of Propo-sition I.6.Finally,in Section VI we make some remarks characterizing so-called“practical”ISS stability in terms of ISS stability with respect to compact sets.II.Some Simple ImplicationsWe start with the proof of Proposition I.1.Proof:We will show the following property,which is equivalent to(LS):∀ε>0∃δ>0st:∀|ξ|A≤δ∀ u ∞≤δsupt≥0|x(t,ξ,u)|A≤ε.(16)Indeed,assume givenε>0.Let T=T(ε/2,1).Pick anyδ1>0so thatγ(δ1)<ε/2.Then:for all|ξ|A≤1and u ∞≤δ1supt≥T|x(t,ξ,u)|A≤ε/2+γ( u ∞)<ε.(17)By continuity(at u≡0and states in A)of solutions with respect to controls and initial conditions,and compactness and zero-invariance of A,there is also someδ2=δ2(ε,T)> 0so that|η|A≤δ2and u ∞≤δ2⇒supt∈[0,T]|x(t,η,u)|A≤ε.Together with(17),this gives the desired property with δ:=min{1,δ1,δ2}.We now prove Lemma I.2.Proof:Weﬁrst note that the0-global asymptotic sta-bility property with respect to A implies the existence of a smooth function V such thatα1(|ξ|A)≤V(ξ)≤α2(|ξ|A)∀ξ∈R n,for someα1,α2∈K∞,and∇V(ξ)f(ξ,0)≤−α3(|ξ|A)∀ξ∈R n,for someα3∈K∞(this is well-known;see for instance,[9] for one such a converse Lyapunov theorem).Following ex-actly the same steps as in the proof of Lemma3.2in[13], one can show that there exists some functionχ∈K∞such that for allχ(|v|)≤|ξ|A≤1,∇V(ξ)f(ξ,v)≤−α3(|ξ|A)/2.(18) (Here we note that in the proof of Lemma3.2in[13],the function g(s)=1for s∈[0,1].)Using exactly the same arguments used on page441 of[12],one can show that there exist a KL-functionβand a K∞-functionγso that if|x(t,ξ,u)|A≤1for all t∈[0,T) for some T>0,then it holds that|x(t,ξ,u)|A≤max{β(|ξ|A,t),γ( u ∞)}(19) for all t∈[0,T).Letρ=min{κ−1(1/2),γ−1(1/2)},where κ(r)=β(r,0)for r≥0.Note here thatρ≤κ−1(1/2)≤1/2.We now show that the(LISS)property holds with theseβ,γ,andρ.Fix anyξand u with|ξ|A≤ρandu ∞≤ρ.First note that|x(t,ξ,u)|<1for t small enough.Claim:|x(t,ξ,u)|A≤1for all t≥0.Assume the claim is false.Then witht1=inf{t:|x(t,ξ,u)|A≥1},it holds that0<t1<∞.Note then that|x(t,ξ,u)|A<1 for all t∈[0,t1).This then implies that|x(t,ξ,u)|A≤max{β(ρ,0),γ(ρ)}≤1/2∀t∈[0,t1). By continuity,|x(t1,ξ,u)|A<1,contradicting to the deﬁ-nition of t1.This shows that t1=∞,i.e.,|x(t,ξ,u)|A≤1 for all t≥0.Thus the estimate in(19)holds for all t,as desired.Next we prove Lemma I.3:boundedness property with respect to A and local stability property with respect to A implies global stability property with respect to A(the converse is obvious).Proof:Assume that Equations(BND)and(LS)hold, for a given choice ofδ,σ1,σ2,α1,α2.Pick a constant c≥0and two class-K functionsβ1andβ2so that,for each i=1,2,σi(s)≤βi(s)+c for all s≥0.Pick two class-K functionsγ1andγ2so that,for each i=1,2,it holds that:γi(s)≥αi(s)∀0≤s≤δ,γi(s)≥2βi(s)∀s≥0,γi(s)≥2[βi(s)+2c]∀s≥δ.Consider anyξand u.Then Equation(GS)holds.In-deed,if both|ξ|A≤δand u ∞≤δthen this follows from Equation(LS).Assume now that|ξ|A>δ.Thus Equation(BND)implies that,for all t≥0,|x(t,ξ,u)|A≤σ1(|ξ|A)+σ2( u ∞)≤β1(|ξ|A)+c+β2( u ∞)+c≤β1(|ξ|A)+2c+(1/2)γ2( u ∞)≤(1/2)[γ1(|ξ|A)+γ2( u ∞)]≤max{γ1(|ξ|A),γ2( u ∞)}.The case u ∞>δis similar.Lemma I.4says that the limit property with respect to A plus the global stability property with respect to A imply the asymptotic gain property with respect to A;it is shown as follows.Proof:Letσ1,σ2,γ∈Nbe as in(LIM)and(GS). We claim that(AG)holds with:γ(s):=max{(σ1◦γ)(s),σ2(s)}.Pick anyξ,u,and anyε>0.By(LIM),there is some T≥0so that|x(T,ξ,u)|A≤γ( u ∞)+ε.Applying(GS) to the initial state x(T,ξ,u)and the control v(t):=u(t+T) we conclude thatlim t→+∞|x(t,ξ,u)|A≤supt≥T|x(t,ξ,u)|A≤max{σ1(γ( u ∞)+ε),σ2( u ∞)}and takingε→0provides the conclusion.Finally,we show:Lemma II.1:Property(14)is equivalent to(AG).Proof:Sinceγ(limt→+∞|u(t)|)≤γ( u ∞),Property(14)implies(AG),with the sameγ.Conversely,assume that(AG)holds;we next show that Property(14)holds withthe sameγ.Pick anyξ∈R n,control u,andε>0.Letr:=limt→+∞|u(t)|.Let h>0be such thatγ(r+h)−γ(r)<ε.Pick T>0so that|u(t)|≤r+h for almost all t≥T,andconsider the functions z(t):=x(t+T)and v(t):=u(t+T)deﬁned on R≥0.Note that v is a control with v ∞≤r+hand that z(t)=x(t,ζ,v),whereζ=x(T,ξ,u).By thedeﬁnition of the asymptotic gain property with respect toA,applied with initial stateζand control v,limt→+∞|x(t,ξ,u)|A=limt→+∞|z(t,ζ,v)|A≤γ( v ∞)≤γ(r+h)<γ(r)+ε.Lettingε→0gives Property(14).III.Uniform Reachability TimeLet(1)be a forward-complete system.For each subsetO of the input-value space U,each T≥0,and each subsetC⊆R n,we denoteR T O(C):={x(t,ξ,u)|0≤t≤T,u∈M O,ξ∈C}andR O(C):={x(t,ξ,u)|t≥0,u∈M O,ξ∈C}=T≥0R T O(C).In[9,Proposition5.1],it is shown that:Fact III.1:Let(1)be a forward-complete system.Foreach bounded subset O of the input-value space U,eachT≥0,and each bounded subset C⊆R n,R T O(C)isbounded.PGiven aﬁxed system(1)which is forward-complete,apointξ∈R n,a subset S⊆R n,and a control u,one mayconsider the“ﬁrst crossing time”τ(ξ,S,u):=inf{t≥0|x(t,ξ,u)∈S}with the convention thatτ(ξ,S,u)=+∞if x(t,ξ,u)∈Sfor all t≥0.The following result and its corollary are central.Theystate in essence that,for bounded controls,ifτ(ξ,S,u)isﬁnite for all u then this quantity is uniformly bounded overu,up to small perturbations ofξand S,and(the Corollary)uniformly on compact sets of initial states as well.(Observethat we are not making the assumption that f is convex oncontrol values and that the set of such values is compactand convex,which would make the result far simpler,bymeans of a routine weak- compactness argument.)Theresult will be mainly applied in the following special case:O is a closed ball in R m,W=R n,and for a given compactset A,C(in the Corollary)is a closed ball of the typeB (A ,2s ),p ∈C ,Ω=B (A ,2s ),and K =B (A ,(3/2)s ).But the general case is not harder to prove,and it is of independent interest.Lemma III.2:Let (1)be a forward-complete system.As-sume given:•an open subset Ωof the state-space R n,•a compact subset K ⊂Ω,•a bounded subset O of the input-value space U ,•a point p ∈R n,and •a neighborhood W of p ,so thatsup u ∈M Oτ(p,Ω,u )=+∞.(20)Then there is some point q ∈W and some v ∈M O suchthatτ(q,K,v )=+∞.(21)Proof:Let p 0=p be as in the hypotheses.Thus for each integer k ≥1we may pick some d k ∈M O so that x (t,p 0,d k )∈Ωfor all 0≤t ≤k .For each j ≥1,we let θj (t )=x (t,p 0,d j ),t ≥0.Consider ﬁrst {θj (t )}j ≥1as a sequence of functions de-ﬁned on [0,1].From Fact III.1we know that there exists some compact subset S 1of R n such that x (t,p 0,d j )∈S 1for all 0≤t ≤1,for all j ≥1.Let M =sup {|f (ξ,λ)|:ξ∈S 1,λ∈O}.Then d θj (t ) ≤M for all j and almostall 0≤t ≤1.Thus the sequence {θj (t )}j ≥1is uniformly bounded and equicontinuous on [0,1],so by the Arzela-Ascoli Theorem,we may pick a subsequence {σ1(j )}j ≥1of {j }j ≥1with the property that {θσ1(j )(t )}j ≥1converges to a continuous function κ1(t ),uniformly on [0,1].Now we consider {θσ1(j )(t )}j ≥1as a sequence of functions deﬁned on [1,2].Using the same argument as above,one proves that there exists a subsequence {σ2(j )}j ≥1of {σ1(j )}j ≥1such that {θσ2(j )(t )}j ≥1converges uniformly to a func-tion κ2(t )for t ∈[1,2].Since {σ2(j )}is a subsequence of {σ1(j )},it follows that κ2(1)=κ1(1).Repeating the above procedure,one obtains inductively on k ≥1a subse-quence {σk +1(j )}j ≥1of {σk (j )}j ≥1such that the sequence {θσk +1(j )(t )}j ≥1converges uniformly to a continuous func-tion κk +1on [k,k +1].Clearly,κk (k )=κk +1(k )for all k ≥1.Let κbe the continuous function deﬁned by κ(t )=κk (t )for t ∈[k −1,k )for each k ≥1.Then on each interval [k −1,k ],κ(t )is the uniform limit of {θσk (j )(t )}.Since the complement of Ωis closed and the θj ’s have images there,it is clear that κremains outside Ω,and hence outside K .If κwould be a trajectory of the system corresponding to some control v ,the result would be proved (with q =p 0).The diﬃculty lies,of course,in the fact that there is no reason for κto be a trajectory.However,κcan be well approximated by trajectories,and the rest of the proof consists of carrying out such an approximation.Some more notations are needed.For each control d with values in O ,we will denote by ∆d the control given by ∆d (t )=d (t +1)for each t in the domain of d (so,for instance,the domain of ∆d is [−1,+∞)if the domain of d was R ≥0).We will also consider iterates of the ∆operator,∆k d ,corresponding to a shift by k .Since K is compact and Ωis open,we may pick an r >0such thatB (K,r )⊆Ω.We pick an r 0smaller than r/2and so that the closed ball of radius r 0around p 0is included in the neighborhood W in which q must be found.Finally,let p k =κ(k )for each k ≥1.Observe that both p 0and p 1are in S 1by construction.Next,for each j ≥1,we wish to study the trajectory x (−t,p 1,∆d σ1(j ))for t ∈[0,1].This may be a priori un-deﬁned for all such t .However,since S 1is compact,wemay pick another compact set S1containing B (S 1,r )in its interior,and we may also pick a function f:R n ×R m →R n which is equal to f for all (x,u )∈ S1×O and has compact support;now the system ˙x = f(x,u )is complete,meaning that solutions exist for all t ∈(−∞,∞).We use x (t,ξ,u )to denote solutions of this new system.Observe that foreach trajectory x (t,ξ,u )which remains in S1,x (t,ξ,u )is also deﬁned and coincides with x (t,ξ,u ).In particu-lar, x (−t,θσ1(j )(1),∆d σ1(j ))=x (−t,θσ1(j )(1),∆d σ1(j )),for each j ,since these both equal x (1−t,p 0,d σ1(j )),for each t ∈[0,1].The set of states reached from S 1,using the modiﬁed system,in negative times t ∈[−1,0],is in-cluded in some compact set (because the modiﬁed system is complete,and again appealing to Fact III.1).Thus,by Gronwall’s estimate,there is some L ≥0so that,for all j ≥1and all t ∈[0,1],x (−t,p 1,∆d σ1(j ))−x (−t,θσ1(j )(1),∆d σ1(j ))≤L p 1−θσ1(j )(1) ,(no “∼”needed in the second solution,since it is also a solution of the original system).Since θσ1(j )(1)→p 1,it follows that there exists some j 1such that for all j ≥j 1,x (−t,p 1,∆d σ1(j ))−x (−t,θσ1(j )(1),∆d σ1(j )) <r 02(22)for all t ∈[0,1].Note that this means in particular thatx (−t,p 1,∆d σ1(j ))∈B (S 1,r/4)⊆ S1for all such t ,for all j ≥j 1,so “∼”can be dropped in Equation (22)for all j ≥j 1.Now let 0<r 1<r 0be such thatx (−t,p,∆d σ1(j 1))−x (−t,p 1,∆d σ1(j 1)) <r 02(23)for all t ∈[0,1],for all p ∈B (p 1,r 1).As this impliesin particular that x (−t,p,∆d σ1(j 1))∈B (S 1,r/2)⊆ S1,again tildes can be bining (22)and (23),it follows that for each p ∈B (p 1,r 1),x (−t,p,∆d σ1(j 1))is deﬁned for all t ∈[0,1]andx (−t,p,∆d σ1(j 1))−x (−t,θσ1(j 1)(1),∆d σ1(j 1)) <r 0(24)for all t ∈[0,1].Let w 1(t )=d σ1(j 1)(t ).Then (24)implies that for each p ∈B (p 1,r 1)it holds that x (−1,p 1,∆w 1)∈B (p 0,r 0),and,since x (−t,θσ1(j 1),∆d σ1(j 1))∈Ωfor all t ∈[0,1],x (−t,p,∆w 1)∈B (K,r/2)∀t ∈[0,1].In what follows we will prove,by induction,that for each i ≥1,there exist 0<r i <r i −1and w i of the form。

伪双曲方程的新混合有限元方法英文

Key wor ds : Pseudo2hyperbolic equations ; H1 2 Galerki n e xpanded mixed fi nit e el e2 ment met hod ; Semidi scret e and full y discret e schemes ; Error e sti ma tes CL C Number :O242. 21 AMS( 2000) Subject Cla ssif ica t ion :65N30 ;65M60 Document code :A Ar t icle ID :100129847 (2010) 0120150208
finite element a nd H1 2 Galer kin mixed finite element me thod is co nstr ucted a nd a nalyze d fo r a class of second or der pseudo2hyper bolic equations. The new for mulatio n not only keeps the a dva ntage s of expa nded mixed for mulatio n but also keep s the advanta ges of H1 2 Galer kin mixed for mula tion. The new mixed for mulation e xpa nds the standar d mixed fo rmulation in the sense that three variables are explicitly t reated :the scala r unknown ,its gradient and it s flux (coefficient times the gra die nt) ,wha t ’ s mor e ,the p ropo sed approac h does not require LBB consiste ncy condition.

Abstract

Fibring Labelled Deduction SystemsJo˜a o Rasga,Am´ılcar Sernadas,Cristina SernadasCMA,Departamento de Matem´a tica,ISTAv.Rovisco Pais1049-001Lisbon,Portugalhttp://www.cs.math.ist.utl.pt/s84.www/cs/{jfr,acs,css}.html{jfr,acs,css}@math.ist.utl.ptLuca Vigan`oInstitut f¨u r InformatikAlbert-Ludwigs-Universit¨a t FreiburgGeorges-K¨o hler-Allee52D-79110Freiburg,Germanyrmatik.uni-freiburg.de/~lucaluca@informatik.uni-freiburg.deAbstractWe give a categorial characterization of how labelled deduction systems for logics with a propositional basis behave under unconstrainedﬁbring and underﬁbring that is constrained by symbol sharing.At the semantic level,we introduce a general semantics for our systems and then give a categorial characterization ofﬁbring of models.Based on this,we establish the conditions under which our systems are sound and complete with respect to the general semantics for the corresponding logics,and establish requirements on logics and systems so that completeness is preserved by both forms ofﬁbring.Keywords:Fibring of Logics,Labelled Deduction Systems,Natural Deduction,Gen-eral Semantics,Category Theory.1IntroductionContext The problem of combining logics has attracted much attention lately.Be-sides leading to interesting applications whenever it is necessary to work with dif-ferent logics at the same time,combining logics is also interesting on purely the-oretical grounds[7],and among the diﬀerent techniques for combining logics,ﬁb-ring[15,17,18,25,26,29]deserves close study.Theﬁbring of two logics is obtained by combining their languages,deduction systems and semantics.The language of theﬁbring is obtained by allowing the use of the language constructors(atomic symbols and connectives)from the logics that we want to combine.So,for example,whenﬁbring a temporal logic and a deontic logic,“mixed”formulae,like((Gα)⇒(O(Fβ))),appear in the resulting logic.In manycases one wants to share some of the symbols,and we then speak ofﬁbring constrainedby symbol sharing.For example,the formula above illustrates the constrained form ofﬁbring imposed by sharing the common propositional part of a temporal and a deonticlogic.Alternatively,we may consider unconstrainedﬁbring,where no symbols of thetwo given logics are shared.The deduction system of theﬁbring is obtained by the free use of the inferencerules from both given logics,provided that these logics are endowed with deductionsystems of the same kind(e.g.both logics with Hilbert systems or both with naturaldeduction systems).Moreover,this approach is of practical interest only if the twogiven deduction systems are schematic,in the sense that their inference rules are“open”for application to formulae with“foreign”symbols.For instance,when insome Hilbert system modus ponens is deﬁned by the rule MP,{(ξ1⇒ξ2),ξ1} ξ2, one may implicitly assume that the instantiation of the metavariablesξ1andξ2byany formulae,possibly with symbols from both logics,is allowed when applying MPin theﬁbring.The semantics of theﬁbring is complicated and,similar to deduction systems,it is better to consider the case where both logics have semantics of the same kind(i.e.with similar models).A possible,quite general,model for many logics with apropositional basis is provided by a triple U,B,ν where U is a set(of points,worlds,states,whatever),B⊆℘U,andν(c):B n→B for each language constructor c of arity n≥0.Given two logics L and L with such a semantics,the semantics of their ﬁbring is a class of models of the form U,B,ν ,such that at each point u∈U it is possible to extract a model from L and one from L .If symbols are shared,the two extracted models should agree on them.This paper follows a program started by[25],where a categorial characterization ofﬁbring of logics with a propositional basis is presented,continued in[26]with the studyofﬁbring of logics with terms and binding operators,and in[29]where preservationof(strong)completeness byﬁbring is obtained for logics with a propositional basis.All these works consider Hilbert-style deduction systems.Here,we concentrate ourattention onﬁbring of logics endowed with labelled deduction systems.To illustrate why considering these systems is important,observe that Hilbert-stylesystems,although uniform,are diﬃcult to use in practice,especially in comparisonwith the more“natural”Gentzen-style systems such as natural deduction,sequentand tableaux systems.Unfortunately,devising Gentzen-style systems for modal,rel-evance and other non-classical logics often requires considerable ingenuity,as well astrading uniformity for simplicity and usability.A solution to this problem is to em-ploy labelling techniques,which provide a general framework for presenting diﬀerentlogics in a uniform way in terms of Gentzen-style deduction systems.The intuitionbehind these techniques is that labelling(sometimes also called preﬁxing,annotatingor subscripting)allows one to explicitly encode additional information,of a semanticor proof-theoretical nature,that is otherwise implicit in the logic one wants to cap-ture.So,for instance,instead of considering a modal formulaϕ,we can consider thelabelled formula w:ϕ,which intuitively means thatϕholds at the point denoted byw in U within the underlying semantics.We can also use labels to specify the wayin which points are related in a particular model;for example,the relational formulaw:R w tells us that the world w is accessible from w in a Kripke model(which is aparticular,simple,instance of the general models of the form U,B,ν ).Labelled deduction systems have been given for several non-classical logics,e.g.[1, 2,3,4,8,10,12,14,16,19,28].Research on labelling has focused not only on the design of systems for speciﬁc logics,but also,more generally,on the characteriza-tion of the classes of logics that can be formalized this way.General properties and limitations of labelling techniques have also been investigated.For example,[4,28] highlight a tradeoﬀbetween limitations and properties:if we reason on the semantic information provided by labelling using Horn-style rules,then we are able to present only a subset of all possible non-classical logics,but we can still capture many of the most common ones and,more importantly,labelling provides an eﬃcient gen-eral method for establishing the metatheoretical properties of these logics,including their completeness,decidability and computational complexity.Moreover,[4,28]also contain some preliminary results on combining labelled deduction systems. Contributions The main contribution of this paper is an account of how labelled deduction systems can beﬁbred.We concentrate on logics with a propositional basis, endowed with labelled deduction systems,and where labels provide additional infor-mation of a semantic nature.Speciﬁcally,our contributions can be summarized as follows.Our approach is general and we give here a novel(algebraic)presentation of labelled deduction systems that provides a suitable basis for deﬁning theirﬁbring,and that subsumes,as simple special cases,the more standard labelled natural deduction systems of our previous work[3,4,28].Based on this presentation,we then give a categorial deﬁnition of(unconstrained and constrained by symbol sharing)ﬁbring of labelled deduction systems.Afterwards,we adapt the general semantics advocated for Hilbert systems in[29]to the case of labelled deduction systems;hence,the gen-eral approach that we consider here subsumes our previous work also at the semantic level(the Kripke-style semantics that we gave for labelled natural deduction systems is a simple,special case of the general semantics).Then we establish the conditions under which our systems are sound and complete with respect to the general seman-tics for the corresponding logics,and deﬁne(both forms of)ﬁbring at the semantic level.Finally,building on the completeness theorem,we establish requirements on the given logics and systems so that completeness is preserved by(both forms of)ﬁbring. Preservation of other metatheoretical properties such as theﬁnite model property and decidability will be subject of future work.Organization The remainder of this paper is organized as follows.In Section2we introduce labelled deduction systems at the abstract level we need and theirﬁbrings. In Section3we focus on semantics,and in Section4we prove the completeness theorem and the preservation results.In Section5we draw conclusions and discuss related and future work.2Labelled deduction systemsIn this section we introduce our labelled deduction systems for logics with a proposi-tional basis.In Section2.1we deﬁne the language of our systems,specifying how to build formulae and judgements.Then,leading up to the deﬁnition of consequence,inSection2.2we deﬁne inference rules and derivations in our systems,giving in passing some examples for modal and temporal logics.Finally,in Section2.3we introduce unconstrained and constrainedﬁbring as categorial constructions and illustrate them with an example from modal logic.2.1LanguageWe introduce the notions of signature,formula and judgement.Deﬁnition1A signature is a pairΣ= C,O of families of language constructors, with C={C k}k∈N where each C k is a set,and O={O k}k∈N+where each O k is a set.The elements of C k are formula operators(or connectives)of arity k,and the elements of O k are relational operators of arity k.Notation2Weﬁx once and for all a family of schema variablesΞ={Ξj}j∈{lb,wf}, whereΞlb is a set of label schema variables andΞwf is a set of(well-formed)formula schema variables.Formulae in our systems are then built fromΣandΞaccording to the following deﬁnition.Deﬁnition3The set L(C,Ξwf)of schema formulae is deﬁned inductively as fol-lows:(i)ξ∈L(C,Ξwf)for allξ∈Ξwf;(ii)c∈L(C,Ξwf)for all c∈C0;and (iii)c(φ1,...,φk)∈L(C,Ξwf)for all c∈C k andφ1,...,φk∈L(C,Ξwf).The set L(O,Ξlb)of relational schema formulae is deﬁned as the set of o(w1,...,w k) for all o∈O k and w1,...,w k∈Ξlb.The set L(Σ,Ξ)of labelled schema formulae is L(Σ,Ξ)={w:ϕ|w∈Ξlb andϕ∈L(C,Ξwf)∪L(O,Ξlb)}. Notation4We write lb(∆)to denote the set of all labels(label schema variables) that appear in the formulae of a set∆⊆L(Σ,Ξ)of labelled schema formulae.The examples below,in particular Examples10and11,will illustrate the idea behind these deﬁnitions:the connectives are in a many-to-many relationship with the relational operators.This is a generalization of the techniques adopted in[3,4,28], where labelled natural deduction systems for several non-classical logics are given by associating each non-classical connective of arity n with a syntactic representative of a semantic relation of arity n+1.1To illustrate this,take the simple example of modal logics.We can evaluate formulae built using the unary modal connectives 2and3in terms of two distinct binary semantic relations R2and R3,which we represent in a modal labelled natural deduction system by means of formulae of the form w:R2w and w:R3w .Moreover,in the case that the logic we are considering is, for example,the basic mono-modal logic K,we have that R2and R3coincide so that in our system for that logic the connectives2and3are associated with the same binary relational operator R.Similar observations apply for the other non-classical logics considered in[3,4,28],e.g.binary relevant implication is associated with a ternary“compossibility”relation.1Note that the techniques of[3,4,28]are in turn an adaptation and extension of the principles of“Boolean algebra with operators”[22]and of“gaggle theory”[13].In[3,4,28]we restricted our attention to relationships between connectives and relational operators that are bijective relationships and where the arities areﬁxed beforehand.But there are logics in which this is not the case;a notable example is given by temporal logics,where the binary“until”connective is associated with a relation that is also binary(cf.Example11).Hence,here we loosen these restrictions: we allow for a liberal relationship between connectives and relational operators,and we do notﬁx their arities beforehand.There is however an aspect in which we follow the restrictions of previous work more closely,and we do so in order to simplify the development.Deﬁnition3says that in our systems we consider two distinct kinds of unlabelled schema formulae,so that the meaning of a labelled schema formula w:ϕwill explicitly depend onϕ.If ϕis a schema formula c(φ1,...,φk),then w:ϕmeans that the formula c(φ1,...,φk) holds at the world denoted by w.Ifϕis a relational schema formula o(w1,...,w k), then w:ϕmeans that the worlds denoted by w1,...,w k are in the relation denoted by o at the world denoted by w.For example,when o is the accessibility relation R of modal logics,w1:R w2means that the expression“w2is accessible”,denoted by R w2,holds at w1,i.e.that at w1we can see w2.We do not admit“hybrid”formulae where the operators can be freely mixed, e.g.c1(o1(c2(w1,w2))).Formulae of this form are used for instance in presentations of non-classical logics based on hybrid languages,e.g.[6],or on semantic embeddings (i.e.on translations intoﬁrst or higher-order logic;see,for example,[23]).Moreover, our relational formulae are atomic and cannot be composed.A consequence of these restrictions is that we will not be able to give labelled deduction systems for all logics that can be presented using hybrid languages,semantic embeddings or other similar approaches based on labelling or translations.However, like for[3,4,28],to which the reader is referred for further details and comparison with related approaches,the restricted language that we adopt will suﬃce to give labelled deduction systems for several of the most common non-classical logics;in fact,thanks to our generalizations,we will be able to give labelled deduction systems for many more logics than the ones considered in previous work.2Returning to the technical development,we now deﬁne what is a judgement. Loosely speaking,a judgement is a relation between a labelled schema formulaδ, aﬁnite set of labelled schema formulae∆,and aﬁnite set of label schema variables Φ,which will allow us to deﬁne when a formula follows from a set of formulae(cf.Def-initions21and22),and also to represent intermediate steps in a derivation where fresh variables may occur.Deﬁnition5A judgement ∆,δ,Φ is a triple where∆∈℘ﬁn L(Σ,Ξ)is aﬁnite set of labelled schema formulae,δ∈L(Σ,Ξ)is a labelled schema formula,andΦ⊆Ξlb is a ﬁnite set of label schema variables,which will represent fresh variables in a derivation.Notation6We write J(Σ,Ξ)to denote the set of all judgements.Moreover,we 2It is then interesting to investigate if our generalizations allow us to inherit the properties shown for the labelled natural deduction systems of previous work,e.g.normalization of derivations and the subformula property.We conjecture that this is the case,but leave a detailed analysis for future work.represent a judgementζ= ∆,δ,{w1,...,w k} by∆,δ w1,...,w k,and we also denote∆by Ant(ζ),δby Cons(ζ),and{w1,...,w k}by Fresh(ζ).Fur-thermore,when Fresh(ζ)is empty,we representζby writing simply ∆,δ .2.2Deduction and consequenceIn order to deﬁne a consequence relation for proving assertions using hypotheses, which we will do in Prop/Deﬁnition27,we need to say when a judgement follows (i.e.can be derived)from a set of judgements.Hence,we now introduce inference rules and deﬁne our labelled deduction systems.Deﬁnition7An inference rule is a pair Prem,Conc where Prem∈℘ﬁn J(Σ,Ξ)is aﬁnite set of premise-judgements,and the conclusion-judgement Conc∈J(Σ,Ξ)is such that Ant(Conc)=∅and Fresh(Conc)=∅. Notation8We represent an inference rule r= {ζ1,...,ζk},ζ byζ1···ζkr,ζand we denote the premises{ζ1,...,ζk}by Prem(r)and the conclusionζby Conc(r). We write R(Σ,Ξ)to denote the set of all inference rules. Deﬁnition9A labelled deduction system(lds)is a pair Σ,D whereΣis a signature and D⊆R(Σ,Ξ).Even though we imposed some simplifying restrictions on signatures,e.g.that schema formulae cannot be mixed with(atomic)relational schema formulae,our inference rules and lds’s are general enough to subsume(as simple instances)the labelled natural deduction systems of[3,4,28]for modal,relevance and other non-classical logics.3To illustrate this,we now give lds’s for the mono-modal logic K and its extensions, and for basic temporal logic with⊥,→and the until operator U.In the case of modal logics,we can give lds’s that are indeed simple notational variants of the labelled natural deduction systems of previous work,and which contain elimination and introduction rules for each connective c(denoted by c E and c I),except for⊥,for which we give only the rule⊥E.3By natural deduction systems we mean,as is usual,systems for proof under assumption that consist of introduction and elimination rules for each of the connectives except falsum,i.e.⊥,for which only an elimination rule is given[27].These rules deﬁne the meaning of each connective, specifying how it is introduced in a formula or eliminated from it.Our lds’s subsume labelled natural deduction systems because we do not commit ourselves here to such requirements on the rules but allow for the more general form of rules of Deﬁnition7.For example,while we do require that our rules are single-conclusioned,we do not require that they necessarily have the form that makes them introduction or elimination rules.Related to this,it is interesting to investigate whether our general approach generalizes,in a similar way,other forms of labelled Gentzen-style systems such as labelled tableaux or sequent systems;we conjecture that,with minor modiﬁcations,this is indeed the case but leave a detailed analysis for future work.Example10An lds Σ,D for the mono-modal logic K is deﬁned as follows:Σ= C,O is such that C0=Π∪{⊥}whereΠis some set of propositional symbols, C1={2},C2={→},and O1={R};D contains the rules:{w:ξ→⊥},w :⊥∅,w:ξ ⊥E, ∅,w:ξ1→ξ2 ∅,w:ξ1∅,w:ξ2 →E, {w:ξ1},w:ξ2∅,w:ξ1→ξ2 →I,∅,w:2ξ ∅,w:R w∅,w :ξ 2E, {w:R w },w :ξ w∅,w:2ξ 2I.By considering the rule2I(2-Introduction),we can now informally explain the role of theﬁnite set of fresh label schema variables.(We will give a formal expla-nation after Deﬁnition19below.)Intuitively,the rule2I says that if for every w accessible from w we haveξ(i.e.w :ξfollows from w:R w with w fresh),then wecan conclude that in w we have2ξ(i.e.w:2ξ).In other words,the universal quan-tiﬁcation corresponds to the side condition that w is fresh,and the antecedent of the premise-judgement requires that w is accessible from w.Note also that,given C,we can deﬁne other formula operators using standard abbreviations,e.g.¬ξ≡abrv(ξ→⊥),ξ1∧ξ2≡abrv¬(ξ1→¬ξ2),and3ξ≡abrv¬2¬ξ. The inference rules for these connectives are then derived from those for the primitive ones in C,as we show for instance in Example24for3elimination.Systems for other modal logics extending mono-modal K are obtained by extending the lds for K with rules expressing properties of R.For example,we obtain lds’s for T and K4by respectively extending the lds for K with the rules∅,w:R w reﬂand ∅,w:R w ∅,w :R w∅,w:R w trans,we obtain an lds for S4by extending the lds for K with both reﬂand trans,and an lds for S5by further adding a rule for the Euclidean property,e.g.∅,w:R w ∅,w:R w∅,w :R w eucl.More generally,we can extend our lds’s with Horn-style rules similar to the“relational rules”of[3,4,28].That is,with rules of the form∅,w1:ϕ1 ··· ∅,w n:ϕn∅,w:ϕwhereϕi is a relational schema formula for each i such that0≤i≤n.Since these example labelled deduction systems are not much diﬀerent from the ones in[3,4,28],we refer the reader to these works for further examples of systems for modal and other non-classical logics,most notably relevance logics.4The following 4Note that the guarantee that our lds’s for K and its extensions are indeed deduction systems for these logics,i.e.that they are sound and complete with respect to the standard Kripke semantics for these logics,follows then trivially by the soundness and completeness of the labelled natural deduction systems of previous work.example,on the other hand,shows that the general approach taken here allows us to give a labelled(natural)deduction system for basic temporal logic,which was not possible with the formal machinery of previous work.Example11An lds Σ,D for basic temporal logic(with⊥,→,and the until opera-tor U associated with a binary relational operator R)is deﬁned as follows:Σ= C,O is such that C0=Π∪{⊥}whereΠis some set of propositional symbols,C2={→,U} and O1={R};D contains rules⊥E,→E,→I,trans,and the following rules for U: ∅,w:ξ1Uξ2 ∅,w:R w {w:R w ,w :ξ2},w :R w w∅,w :ξ1 U E1,∅,w:ξ1Uξ2 {w:R w ,w :ξ2},w :ξ3 w∅,w :ξ3 U E2,∅,w :ξ2 {w:R w ,w :R w },w :ξ1 w∅,w:ξ1Uξ2 U I.Note,however,that due to the restrictions we imposed on our language we are not able to give lds’s for extensions of this logic where time is linear and/or discrete, which would require us to introduce connectives(e.g.disjunction)with which we can compose relational formulae.Moreover,even though here we can give rulesζ1···ζkrζ0in which we mix judgementsζi,where Cons(ζi)is a labelled schema formula,with judgementsζj,where Cons(ζj)is a labelled relational schema formula,there are other logics that require further interaction between formula and relational operators.That is,there are logics that require us explicitly to consider hybrid formulae,in which operators can be mixed freely.5We now motivate the following deﬁnitions.The objective is to say when a formula γis derived from a set of formulaeΓin an lds Σ,D ,i.e.Γ Σ,D γ.We will say that this is the case when the judgement Γ,γ,∅ is derived from the empty set of judgements.For this purpose we have to deﬁneﬁrst what it means that a judgement ζis derived from a set of judgementsΘin an lds Σ,D ,i.e.Θ∼ Σ,D ζ,and we do so after introducing some basic notions such as substitutions,instantiations and derivation trees.Deﬁnition12A substitution is a pairσ= σlb,σwf ,whereσlb:Ξlb→Ξlb and σwf:Ξwf→L(C,Ξwf)are maps called respectively label substitution and formula substitution.5In other words,we allow mixing of operators at the metalevel,by having rules that mix judge-ments with diﬀerent kinds of formulae as conclusions,but we forbid mixing of operators at the object level,i.e.inside formulae.Note also that although the inference rules we give here are more general than the ones in[3,4,28],the above limitations are a consequence of the results reported in these works as,mutatis mutandis,the observations made there apply straightforwardly in this setting.As we remarked above,labelled deduction systems built using an extended language will be subject of future work.Notation13We write Sub(Σ,Ξ)for the set of all substitutions,and id for the identity substitution onΞ.Since our inference rules are schematic,to be able to carry out derivations we need to deﬁne the concepts of substitution for a rule and instantiation.For the former we need to say when two assignments are equivalent up to a particular set of labels. Deﬁnition14Letσandσ be substitutions andΦ⊆Ξlb be a set of labels.We saythatσandσ areΦco-equivalent,in symbolsσ≡Φσ ,iﬀσwf=σ wf andσlb(w)=σ lb(w)for each w∈Ξlb such that w∈Φ. Deﬁnition15Let r= {ζi}1≤i≤k,ζ be an inference rule.An r-substitution is a pair {σi}1≤i≤k,σ such thatσi≡Fresh(ζi)σfor each i=1,...,k. Notation16In the following,given a labelled schema formulaδ,a set of labelled schema formulae∆and a substitutionσ,we denote byδσthe labelled schema formula that results fromδby the simultaneous replacement of its schema variables by their value given byσ.By extension,∆σdenotes the set consisting of allγσfor each γ∈∆.Similarly,given a label w and a set of labelsΦ,we denote by wσthe label σlb(w),and denote byΦσthe set consisting of all wσfor each w∈Φ. Deﬁnition17The judgement Γ,γ,Υ is an instance of the judgement ∆,δ,Φ by the substitutionσiﬀ(i)δσisγ,(ii)Φσ⊆Υ,(iii)∆σ⊆Γ,and(iv)wσ∈(Ξlb\{w})σfor each w∈Φ. Notation18We denote by Inst(ζ,σ)the set of instances of a judgementζbyσ. Deﬁnition19The pair { Γi,γi,Υi }1≤i≤k, Γ,γ,Υ is an instance of r= { ∆i,δi,Φi }1≤i≤k, ∅,δ,∅ by an r-substitution {σi}1≤i≤k,σ iﬀδσisγ,and for each i,with 1≤i≤k,• Γi,γi,Υi ∈Inst( ∆i,δi,Φi ,σi),•lb(Γ)∩Φiσi=∅,•Γi=Γ∪∆iσi,and•Υi\Φiσi=Υ.Observe that,reasoning backwards from goal to axioms,i.e.applying rules from conclusion to premises,the image of any fresh variable in a premise should not have been already used in the derivation we are building,and that the hypotheses in the conclusion are transferred to the premises,and so are(more strictly)the fresh variables.Consider again the rule{w:R w },w :ξ w∅,w:2ξ 2I.The meaning of the“fresh”label schema variable w can now be speciﬁed formally as follows:any instantiation of w is a variable not present in the formulae of the instance of the judgement except in the ones indicated in the judgement under instantiation (i.e.w:R w and w :ξ.)Notation20We denote by Inst(r,s)the set of all rules that are instances of rule r by the r-substitution s. Deﬁnition21A judgementζis derivable from a set of judgementsΘin an lds Σ,D ,in symbolsΘ∼ Σ,D ζ,iﬀthere is a tree annotated by judgements such that(i)the annotation of the root isζ,and(ii)for each node,•the pair formed by the annotation of the node and the set of annotations of its successors is an instance of some rule of D,•or the node has no successors and its annotation is an axiom,i.e.it is of the form Γ∪{γ},γ,Υ ,•or it has no successors and there is an hypothesis ∆,δ,Φ ∈Θand a substitution σ≡Φid such that if its annotation has the form Γ,γ,Υ then∆σ⊆Γ, lb(∆∪{δ})σ∩Υ⊆Φσandγisδσ.A tree in the conditions of the previous deﬁnition will be called a derivation tree forΘ∼ Σ,D ζ.In the following,when there is no ambiguity,we will identify a node with its annotation.When it is not convenient to do so,we will refer to the annotation of a node x by a(x)and even by a T(x)when referring to the underlying tree T.We can nowﬁnally deﬁne what are derivations in our lds’s.Deﬁnition22A labelled schema formulaγis derivable from a set of labelled schema formulaeΓin an lds Σ,D ,in symbolsΓ Σ,D γ,iﬀ∅∼ Σ,D Γ,γ . Notation23When no confusion arises,we will simply writeΘ∼ζandΓ γinstead ofΘ∼ Σ,D ζandΓ Σ,D γ,respectively.As an example,we now employ the derived rules for negation to derive rules for 3in our lds for the mono-modal logic K.All these derived rules are clearly also derivable in lds’s extending that for K.Example24It is easy to see that,in our lds for the mono-modal logic K, ∅,w:⊥ can be derived from ∅,w:¬ξ and ∅,w:ξ ,and that ∅,w:¬ξ can be derived from {w:ξ},w:⊥ .Hence,the following rules for negation are derivable:∅,w:¬ξ ∅,w:ξ∅,w:⊥ ¬E and {w:ξ},w:⊥ ∅,w:¬ξ ¬I.In(1)we derive ∅,w :ξ2 from the hypotheses {w :ξ1,w:R w },w :ξ2 w and ∅,w:3ξ1 .{w :ξ2→⊥},w:¬2¬ξ1 HYP(2){w :ξ2→⊥,w:R w },w :¬ξ1 w ¬I {w :ξ2→⊥},w:2¬ξ1 2I{w:ξ2→⊥},w:⊥ ¬E∅,w :ξ2 ⊥E(1){w:⊥→⊥,w:ξ2→⊥,w:ξ1,w:R w},w:ξ2→⊥ w AX(3)(2){w :⊥→⊥,w :ξ2→⊥,w :ξ1,w:R w },w :⊥ w →E{w :ξ2→⊥,w :ξ1,w:R w },w :⊥ w ⊥E{w :⊥→⊥,w :ξ2→⊥,w :ξ1,w:R w },w :ξ2 w HYP(3) Hence,the following rule for3elimination is derivable:∅,w:3ξ1 {w :ξ1,w:R w },w :ξ2 w∅,w :ξ2 3E.Similarly,we can derive the introduction rule∅,w :ξ ∅,w:R w∅,w:3ξ 3I,since we can easily show that{ ∅,w :ξ , ∅,w:R w }∼ ∅,w:3ξ Two remarks.First,note that3E inherits from2I a freshness side condition on a label schema variable:w is a new variable that is diﬀerent from w and w ,and does not occur in any formula of the premise-judgement other than w:R w and w :ξ. Second,to illustrate the instantiation of rules by given substitutions,note that in the instantiation of2I(with straightforward substitutions)in the derivation(1)the antecedent of the conclusion(w :ξ2→⊥)is inherited in the premise,and that the other formula(w:R w )in the antecedent of the premise is an instantiation of the corresponding formula in the rule.We now prove two useful properties of derivations,which will allow us to introduce a consequence operator in Prop/Deﬁnition27.Weﬁrst show that derivations are monotonic,in the sense that if a labelled schema formulaγis derivable from a set of labelled schema formulaeΓ,thenγcan be derived also fromΓ∪Ψ,whereΨis an arbitrary set of labelled schema formulae.Then,in Lemma26,we show that derivations are idempotent,since if∅∼ Γ,ψ and∅∼ {ψ},γ then∅∼ Γ,γ . Lemma25For eachΨ⊆L(Σ,Ξ)andΥ⊆Ξlb we have∅∼ Γ∪Ψ,γ,Υ whenever ∅∼ Γ,γ .Proof Assuming that we are given a derivation tree for∅∼ Γ,γ ,we want to deﬁne a derivation tree for∅∼ Γ∪Ψ,γ,Υ .The basic idea is to rename the fresh variables used in theﬁrst derivation so that no clash appears with the variables occurring inΨandΥ.This is achieved by introducing a family of substitutions that for each node will carry out the renaming of the variables of the judgement in that node.Consider a derivation tree T for∅∼ Γ,γ ,and for each node x deﬁne a substitu-tionρx such that•the substitution of the node root is id,and。

【吴耀武核心词汇100篇】

I will not give you a fish but tell you how to fish…考研英语词汇全攻略主编: 吴耀武海文考研教育研究中心主讲教师简介吴耀武:著名英语教学与测试专家；西安外国语大学语言测试研究中心副主任；全国公共英语等级考试（PETS）考官；先后两次赴香港岭南大学进修和工作；陕西电视台《考试联盟》栏目、西安音乐台《快乐时光》栏目、西安经济广播电台《空中英语角》节目特邀嘉宾。

现已出版考研英语《黑博士系列丛书》（人民日报出版社）、大学英语四六级《考试专家系列丛书》、《全攻略系列丛书》（西工大出版社）和英语专业四八级《TEM-4/考试指南》等系列丛书（世界图书传版社）30余部，近800万字。

于2001年起在全国高校作关于考研英语、雅思（IELTS）、英语专业四八级（TEM-4/8）和大学英语四六级（CET-4/6）巡回讲座，在各大高校均引起巨大轰动，讲座场场爆满。

并受到新浪、搜狐、中国考试网、都市快报、西安零距离、华商报、西安晚报、三秦都市报等多家媒体的关注和专访。

他在英语培训领域提出了自己独特的教学理念——努力改进学习方法、量身定做学习计划；紧紧把握考试脉搏，传授英语应试策略；奠定良好心理基础，培养进取人生态度。

这一理念在课堂上得以贯彻并为广大学员所认可。

现在吴老师每年在北京、上海、西安、武汉、南京、石家庄、沈阳、天津、乌鲁木齐等20多所中心城市主讲考研英语和大学英语四六级课程，经他直接培训的学员已逾10万人，听众更是不计其数。

学员对他的评价是：――― 感情真挚，敬业专注；――― 才思敏捷，经典幽默；――― 传授知识如春风化雨，润物无声；――― 谈应试技巧如数家珍，挥洒自如；――― 他的语言总是富有人生哲理，他的课堂永远充满着欢笑从容；他传播的不仅仅是英语知识，更是一种人生感悟！第一部分考研英语核心词汇记单词时不要采用从A——Z的顺序，而是要灵活搭配，比如A和J，B和O，C和N，D和L等，这样记起来既不觉得累又有成就感。

_第二十二条军规_中黑色幽默的语言表征探讨[1]

第21卷第1期合肥工业大学学报(社会科学版)V o l.21N o.1 2007年2月JOU RNAL O F H EFE I UN I V ER S IT Y O F T ECHNOLO GY(Social Sciences)Feb.2007《第二十二条军规》中黑色幽默的语言表征探讨张　浩(合肥工业大学人文经济学院,安徽合肥　230009)摘　要:文章旨在探讨约瑟夫・海勒最具代表性的小说《第二十二条军规》中运用黑色幽默手法来服务其荒诞的非理性主题的诸种语言表征。

基于对文本中大量的黑色幽默语言的实例分析,分别从悖论、平行对仗、重复、比照和并置等修辞层面,揭示了作者运用黑色幽默深化小说的非理性主题的手段,并指出黑色幽默的语言特色正是其展示的非理性世界的超现实表征。

关键词:黑色幽默;语言表征;非理性主题;第二十二条军规中图分类号:I106.4 文献标识码:A 文章编号:100823634(2007)0120115204O n the lingu istic rep resen tati on s of b lack hunm o r in Ca tch22ZHAN G H ao(Schoo l of H um anities and Econom ics,H efei U niversity of T echno logy,H efei230009,Ch ina)Abstract:It is w idely acknow ledged that Jo sep h H eller is one of the m o st successfu l rep resen tative w riters of b lack hum o r gen re.T he p u rpo se of th is p ap er is to discu ss how the lingu isitic rep resen tati on s of b lack hum o r ranging from p aradox,p arallelis m,rep etiti on,con trast and jux tapo siti on w ere sk illfu lly em p loyed in the novel Ca tch22to serve its ab su rd them e w ith concrete, typ ical exam p les.B ased upon th is analysis,the p ap er concludes that th is avan t2garde literary techn ique is the m o st effectivelly m an i p u lated literary device to con tribu te to the developm en t of the ab su rd them e in Ca tch22and it is also the lingu istic carrier fo r the su rrealistic rep resen tati on of the lunatic w o rld.Key words:b lack hum o r;the lingu istic rep resen tati on s;irrati onal them e;Ca tch22一、引言1961年出版的《第二十二条军规》,因其对战争、资本主义自由企业竞争和当时腐朽的美国生活方式的有力鞭笞,使得约瑟夫・海勒一跃成为当时文坛的炙手可热的知名作家。

The Intentional Planning System ItPlanS

The Intentional Planning System: ItPlanS
Christopher W. Geib University of Pennsylvania Department of Computer and Information Science 200 S. 33rd Street Philadelphia, PA 19104 Phone: (215)898-8745 Email: geib@
This research is supported by ARO grant no. DAAL03.89.C.0031 PRIME
i
1 Introduction
Consider the action of an agent opening its hand while holding an object. Suppose that the agent performs this action in two di erent situations. In the rst situation, the object in the agent's hand is supported by some other object. In the second situation, the object is not supported. If the agent opens its hand in the rst situation, the object will be stacked on the supporting object. However, if the agent performs the action in the second situation, the object will fall and possibly break. This example highlights the e ect that context can have on action. In many planning systems 6, 11, 13], in order to use the hand opening action for both of these e ects, the system would need to have two separate actions, one for stacking and one for dropping. But creating more actions, will increase the branching factor at each node of the planner's search space. Many planners will have to consider each of the actions, increasing their runtime. This solution also seems to con ict with our intuitions that there is something the same about the stacking action and the dropping action that is not being captured. It is surprising, considering the prevalent nature of conditional e ects of action, that until recently the building of \di erent" actions to accomplish di erent e ects has been seen as a legitimate solution to the dilemma of conditional e ects. This was a result of the fact that early planning systems did not distinguish between an action and its e ects. In fact, any planning system with traditional STRIPS style action representations 3, 8], has this problem. This problem with the traditional STRIPS style representation of actions is not a new observation. In introducing conditional e ects to actions, researchers 9, 12] have acknowledged that context is relevant to determining the e ects of an action, and accepted that actions can have wildly di ering e ects depending on the context of their execution. However their solutions to the problem have been inadequate. One solution 9] has been to add \secondary" preconditions to capture the \conditional" e ects that actions have. Unfortunately this solution is little more than a theoretically sound version of creating separate actions for di erent e ects. Thus, this kind of frame work gives a formal language to talk about conditional e ects of actions but does not provide a practical solution for how to plan using them. It also rejects the fundamental realization of action in context, namely: an action has no e ects outside of the environment in which it is performed. 1

Urbana,IL61801

1 Introduction
In a real-time system, tasks have deadlines. The correctness of the system depends as much on the tasks completing by their deadlines as on the logical correctness of their results. Hence timing predictability is an important issue in the design of a real-time system. Due to the ready availability of inexpensive processors with large memory capacities and communication networks with large bandwidth, distributed computing systems are increasingly being used in many of the real-time applications. The potential speedup o ered by a distributed system not only enhance the performance of the real-time systems, but it also makes some systems, which are otherwise unschedulable, schedulable.
This research is supported in part by ONR under contract N00014-92-J-1815 and NSF under CCR 92-24269.

微细粒矿物的分选技术及设备探讨_尚旭

第1期2007年2月矿产保护与利用C O NSERVAT I O N AND UT I L I ZATI ON O F M I NERAL RE S O URCES№.1Feb.2007微细粒矿物的分选技术及设备探讨*尚旭,张文彬,刘殿文,凌石生,殷志勇(昆明理工大学国土资源工程学院,昆明,650093)摘要:随着矿山开采矿石嵌布粒度的日益变细,微细粒矿物的分选已经成为目前矿物分选的一个重要组成部分,越来越受到人们的关注。

介绍了目前微细粒矿物分选的主要工艺及主要设备,为今后微细粒矿物的分选研究提供参考。

关键词:微细粒矿物;分选;工艺;设备中图分类号:TD923　文献标识码:A　文章编号:1001-0076(2007)01-0031-05The D iscussion on Beneficiation Technologies and Equip m ents for M inutenessM i n eralsSHA NG X u,ZHA NG W e n-bi n,LI U D i a n-wen,et a l.(School of Land Resource Engi n eeri n g,Kunm i n g Un i v ersity of Sc i e nce and Technol o gy,Kun m i n g 650093,C hi n a)Abstract:The bene ficiation o f tiny-fi n e-particle m inerals has been an i m portan t part o fm i n eralprocessing recen tl y because of the triv ialization o f ore size fro m m ines,and peop l e have paid m oreand m o r e attention to i.t The m ain techno l o gies and equ i p m en ts for fine-particle m ine ral process-ing ar e pr esented i n t h is paper.This w ill p r ovide a r e ference for t h eir research in t h e futur e.K ey words:tiny-fine-particle m ine r a l;beneficia tion;techno logy;equip m ent 20世纪初发展起来的浮选技术,曾使矿业发生了一场深刻的革命,使许多品位低、嵌布细的复杂矿石的选矿处理成为现实。

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2. A study of the Fourier transform. In order to show that f (z ) is not 1 an element of the space H 2 (δ Ω), we have to prove the divergence of the following integral:
Because the last term tends to +∞ when n → +∞,
+∞ 1
sin2 (x) dx = x =
1 π
≥
1
+∞ sin2 (x) sin2 (x) dx + dx x x π nπ sin2 (x) sin2 (x) dx + lim dx n→+∞ π x x sin2 (x) 1 1 1 1 dx + lim + + ··· + x 4 n→+∞ 2 4 n
1 0 We know that ft ∈ H 1 (Ω) when t > 1 2 and ft ∈ H (Ω) for t > − 2 , so we might 1 1 is an element of H 2 (δ Ω). However, we prove in this paper that this think that f 2 1 (z ). hypothesis is false. We denote f (z ) = f 2
sin2 (x) dx ≥ x ≥
n−1 k=1 n−1 k=1
(k+1)π kπ
sin2 (x) dx ≥ x
n−1 k=1
n−1 k=1
π kπ + 34
kπ + π 4
sin2 (x) dx x 1 1 1 + + ··· + 2 3 n .
π 1 · = 2(k + 1)π 2
π 1 π
1 1 = 4(k + 1) 4
+∞ 0
(u(x))2 (1 + x2 ) 2 dx,
1
where u(x) is the Fourier transform of f (z ): u(x) =
+∞ −∞ 1
f (z ) e(−2πxzi) dz = 2
0
z 2 cos(2πxz ) dz.
1
∗ Received by the editors June 22, 1995; accepted for publication (in revised form) November 6, 1995. The second author was supported by a fellowship from the Conseller´ ıa de Cultura, Educaci´ oi Ciencia of the Generalitat Valenciana. /journals/sirev/39-3/28678.html † Departamento de An´ alisis Matem´ atico y Matem´ atica Aplicada, Universidad de Alicante, Apartado 99, 03080 Alicante, Spain (villacampa@ua.es). ‡ Departament de Matem` atiques, Universitat Jaume I, Campus Penyeta Roja, 12071 Castell´ on, Spain (balaguer@mat.uji.es).
+∞
2
hence F is 1 (1+ x2 ) 2 dx
sin (x) dx converges. However, the latter integral can be converges if and only if 1 x seen to diverge by the following elementary argument: nπ π
where F (x) =
0
√ 2πx
sin(s2 )ds
for x > 1.
π 8 ), and +∞ (u(x))2 1
The Fresnel integral converges (in fact, limx→+∞ F (x) = bounded. The second and third integrals clearly converge, so
+∞ 1
(u(x))2 (1 + x2 ) 2 dx =
1
1
+∞ +∞ 1
1 4 sin2 (2πx) (1 + x2 ) 2 dx 2 (2πx)
− +
1
8 sin(2πx)F (x) (2πx) 2
5
(1 + x2 ) 2 dx
1
+∞
1 4F (x)2 (1 + x2 ) 2 dx, (2πx)3
494
CLASSROOM NOTES
1
1
495
+∞
(u(x))2 Since 0 (u(x))2 (1+ x2 ) 2 dx exists and is ﬁnite, we need only consider 1 1 ·(1 + x2 ) 2 dx. After one integration by parts and a simple substitution, we obtain the following representation for u(x): √ 2πx 2 sin(2πx) − 2 3 sin(s2 )ds if x > 0, 2πx 0 2 (2 πx ) u(x) = 4 if x = 0. 3 Therefore,
SIAM REV. Vol. 39, No. 3, pp. 494–495, September 1997
c 1997 Society for Industrial and Applied Mathematics
006
A STUDY OF A SEMI-INFINITE INTEGRAL∗
´‡ Y. VILLACAMPA† , A. BALAGUER‡ , AND J. L. USO Abstract. Taking an open and bounded interval in R, Ω which contains the interval [−1, 1], we prove that | x | 2 x ∈ [−1, 1] is not an element of H 2 (δ Ω). The procedure is based on a study of the Fourier transform. Key words. Fourier transform, Sobolev space, semi-inﬁnite integral AMS subject classiﬁcations. A10, B30 PII. S0036144595286786
1 1
1. Introduction. The study of error estimators for elliptic problems with boundary conditions leads to the study of the trace spaces. Such a study comes up when we have an open and bounded interval in R called Ω and an element of the n-order (n ∈ N ) Sobolev space H n (Ω) [3] and we want to deﬁne its value at the boundary of Ω (denoted here by δ Ω). There is a diﬃculty with this prolongation when n ≥ 2 because its elements are not necessarily continuous functions. There exist some techniques that theoretically solve the extension of the trace operator by using the trace spaces, that is, certain Sobolev spaces of real order H s 1 (see [2], [3]). An important trace space is H 2 (δ Ω). An analysis of the functions of 1 H 2 (δ Ω) by the type of discontinuity after extending them, through zero, to all R leads to the study of the family ft , t ∈ R: (1) ft (z ) = | z |t 0 ∀z ∈ [−1, 1], ∀z ∈ (] − ∞, −1[∪]1, +∞[).
= +∞.பைடு நூலகம்
Acknowledgments. We are grateful to the referees for their suggestions and comments.
REFERENCES [1] C. F. GERALD AND P. O. WHEATLEY, Applied Numerical Analysis, Addison–Wesley, Reading, MA, 1989. [2] J. T. ODEN AND J. N. REDDY, An Introduction to the Mathematical Theory of Finite Elements, John Wiley and Sons, New York, 1976. [3] VO-KHAC KHOAN, Distributions. Analyse de Fourier, in Operateur aux Derives Partielles (Tome 1, 2), Ed. Vuibert, 1970.