R. Fuchs' problem of the Painleve equations from the first to the fifth

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The periodic Cauchy problem of the modified Hunter-Saxton equation

a rX iv:mat h /5213v3[mat h.AP]21Nov25The periodic Cauchy problem of the modiﬁed Hunter-Saxton equation Feride Tı˘g lay February 1,2008Abstract We prove that the periodic initial value problem for the modiﬁed Hunter-Saxton equation is locally well-posed for initial data in the space of continuously diﬀerentiable functions on the circle and in Sobolev spaces H s (T )when s >3/2.We also study the analytic regularity (both in space and time variables)of this problem and prove a Cauchy-Kowalevski type theorem.Our approach is to rewrite the equation and derive the estimates which permit application of o.d.e.techniques in Banach spaces.For the analytic regularity we use a contraction argument on an appropriate scale of Banach spaces to obtain analyticity in both time and space variables.In this paper we study the periodic Cauchy problem of the modiﬁed Hunter-Saxton equation ∂t u +u p ∂x u =12(∂x u )2(HS)is called Hunter-Saxton equation.Written in the form (HS)the Hunter-Saxton equation can be viewed as a nonlocal perturbation of the Burgers equation like the Camassa-Holm equa-tion.Both equations Camassa-Holm and Hunter-Saxton arise along with the Korteweg-De Vries equation in these nonlocal forms when derived on the Bott-Virasoro group [KM].Similar local well-posedness results hold for Camassa-Holm equation in Sobolev spaces (see for example the proof given by A.Himonas and G.Misio l ek in [HM2]or [Mis1])and in C 1(the C 1theory for Camassa-Holm equation is developed by G.Misio l ek in [Mis1]).Another context in whichthe three equations Korteweg-De Vries,Camassa-Holm and Hunter-Saxton are treated in a uniﬁed way is their scattering theory described by R.Beals,D.H. Sattinger and J.Szmigielski in[BSS1].All three equations are bihamiltonian and have the same symmetry group,namely Virasoro group as shown in[KM]by B.Khesin and G.Misio l ek.Moreover these equations correspond to the equa-tions of the geodesicﬂow with respect to diﬀerent right invariant Riemannian metrics on this group or on an associated homogeneous space[KM].An observation of V.Arnold[Arn],that the initial value problem for the classical Euler equations of a perfectﬂuid can be stated as a problem ofﬁnding geodesics on the group of volume preserving diﬀeomorphisms,was used by D.G. Ebin and J.Marsden in[EMa]who developed the necessary functional analytic tools and established sharp local well-posedness results for the Euler equations in a class of Sobolev spaces.There is no similar result for p≥2for the modiﬁed Hunter-Saxton equation. It turns out,however,that the method of rewriting the problem as an ordinary diﬀerential equation on the group of diﬀeomorphisms(just as in the case of Euler equations)can be applied by introducing two dependent variables namelyζand γin(1).We use this approach to develop an appropriate analytic framework for(mHS)and prove the following theorems.Theorem1.Let p≥1be any positive integer and s>3/2.Given the initial data u0∈H s(T)the Cauchy problem for the equation(mHS)has a unique solutionu∈C0([0,T),H s(T))∩C1([0,T),H s−1(T))for some T>0and the solution depends continuously on the initial data u0(x).Clearly the local well-posedness in C1(T)as stated in theorem2is a stronger result than the local well-posedness in H s(T)for s>3/2as stated in theorem 1.Theorem2.Let p≥1be any positive integer.For any u0∈C1(T)the Cauchy problem for(mHS)has a unique solutionu∈C0([0,T),C1(T))∩C1([0,T),L∞(T))for some T>0and the solution depends continuously on the initial data.After the completion of this paper we learnt that for p=1theorem1has been proved independently by Z.Yin in[Yi]using the semigroup theory of T. Kato(see the remark2below).Furthermore the two equations,the modiﬁed Hunter-Saxton(mHS)and the Camassa-Holm,do not have similar features only for low regularity data but also for high regularity data.The study of analytic regularity of solutions of the Camassa-Holm equation by A.Himonas and G.Misio l ek[HM1]using an abstract Cauchy-Kowalevski theorem led us to investigate the analytic regularity of the Cauchy problem for(mHS)and prove the following theorem.2Theorem3.If the initial data u0is an analytic function on T then there exists anε>0and a unique solution u of the Cauchy problem for the equation(mHS) that is analytic in both variables x and t on T for all t in(−ε,ε).We would like to note that the analyticity properties of the solutions to the Hunter-Saxton and Camassa-Holm equations are quite diﬀerent from those of the Korteweg-De Vries equation whose solutions are analytic in the space variable for all time[Tru]but are not analytic in the time variable[KaM].Our approach in proving theorem3is to rewrite the equation to use a con-traction argument on an appropriate scale of Banach spaces.This contraction argument was developed in the form of an abstract Cauchy-Kowalevski the-orem by L.V.Ovsjannikov[Ovs1,Ovs2],F.Treves[Tre],L.Nirenberg[Nir], T.Nishida[Nis]and M.S.Baouendi and C.Goulaouic[BG]among others and subsequently applied to the Euler and Navier-Stokes equations.1Well-posedness in C1Many techniques in studying diﬀerential equations are based on Picard’s con-traction argument on Banach spaces.Here we develop the tools to use this argument for the Cauchy problem of(mHS).Let usﬁrst introduce the operator ∂−1x to write the modiﬁed Hunter-Saxton equation in the form(mHS):∂−1x f(x):= x x0f(y)dy− T x x0f(y)dydxfor someﬁxed x0in T.The Hunter-Saxton equation(HS)can be restated on the space D of C1(T) diﬀeomorphisms of T as an ordinary diﬀerential equation.The idea comes from its derivation as a geodesic equation on the Bott-Virasoro group(see[KM]). Similarly one can restate(mHS)as an ordinary diﬀerential equation on the product space D×C1(T)as follows.Letζbe equal to u◦γwhereγis theﬂow generated by u p.Then we obtain the set of equations˙γ=ζp,˙ζ=(∂t (ζ◦γ−1)+(ζ◦γ−1)p∂x(ζ◦γ−1))◦γ.(1)The two initial value problems,for(mHS)and for the equation in(1),are equivalent in the following sense.Proposition1.A function u∈C1(T)is a solution to the Cauchy problem ∂t u+u p∂x u=p2(∂−1x ((ζ◦γ−1)p−1(∂x(ζ◦γ−1))2))◦γ(3)3with initial dataζ(x,0)=u0(x)andγ(x,0)=id x.Therefore it is suﬃcient to prove that the pair(ζp,F(γ,ζ))deﬁnes a con-tinuously diﬀerentiable vectorﬁeld in a neighborhood of(id,0)in D×C1(T). Then theorem2follows by the fundamental theorem on ordinary diﬀerential equations in Banach spaces with the observation that the smooth dependence on initial data in(3)implies continuous dependence on initial data for(2). Proof of Theorem 2.Let us use the convenient notation from[Eb1]and denote by Pγthe operator given by conjugationPγ(g):=P(g◦γ−1)◦γfor anyγ∈D and pseudodiﬀerential operator ing this notation we write the right hand side of theﬁrst equation in(3)asF(γ,ζ)=12∂γ(∂−1x)γg(γ,ζ)+12(∂−1x)γ∂ζg(γ,ζ)(5)and∂γg(γ,ζ)=∂γ(∂x)γ(ζp)(∂x)γζ+(∂x)γ(ζp)∂γ(∂x)γζ.(6) Let s→γs be a smooth curve in D such thatγ0=id and∂sγs|s=0=W for W∈C1(T).By the deﬁnition of the operator∂−1x we obtain∂γ(∂−1x)γ(g)(W)=−(∂−1x)γ(W(∂x)γg)+gW=(∂−1x)γ(g(∂x)γW)+ T gW dx.(7) Similarly we have∂γ(∂x)γ=[W◦γ−1∂x,∂x]γ(8) where the bracket[·,·]denotes the commutator of the operators.By(4)-(8)the directional derivatives of F(γ,ζ)are given by the following explicit formulas:∂γF(γ,ζ)(W)=−12 T W(∂x)γζp(∂x)γζdx,(9)41∂ζF(γ,ζ)(W)=linear operators on C1(T)it suﬃces to prove the continuity of the following mapsγ→∂γ(∂−1x)γ∈L(C1,L(C0,C1))(12)γ→(∂−1x)γ∈L(C0,C1)(13)γ→(∂x)γ∈L(C1,C0)(14) in a neighborhood of(id,0)∈D×C1(T).We consider the map(13)ﬁrst.A change of variables in(∂−1x)γf−∂−1x f= γ(x)γ(x0)f◦γ−1(y)dy− T γ(x)γ(x0)f◦γ−1(y)dy dx− x x0f(y)dy+ T x x0f(y)dy dxleads to the identity(∂−1x)γf−∂−1x f=∂−1x(f(∂xγ−1)).(15) The right hand side of this identity is bounded byCγ f ∞ γ−id C1and therefore we have the estimate(∂−1x)γf−∂−1x f C1≤Cγ f ∞ γ−id C1(16) and the map in(13)is continuous.In order to show the continuity of the map in(12)we observe that the deﬁnite integral in∂γ(∂−1x)γ(W)does not depend on x and thus we have∂γ(∂−1x)γ(W)−∂γ(∂−1x)id(W) C1≤ (∂−1x)γ{(∂x)γζp(∂x)γζ(∂x)γW}−∂−1x(∂xζp∂xζ∂x W) C1(17)+ T W{(∂x)γζp(∂x)γζ−∂xζp∂xζ}dx (18) Adding and subtracting the appropriate terms we can bound(17)byCγ ∂xζp∂xζ∂x W{(∂xγ−1◦γ)3−1} ∞+ (∂−1x)γ{∂xζp∂xζ∂x W((∂xγ−1◦γ)3−1)} ∞(19)+ {(∂−1x)γ−(∂−1x)id}(∂xζp∂xζ∂x W) C1.We use(16)to estimate the third summand above byζ p+1W C1 γ−id C1.C1Clearly theﬁrst two terms in(19)are bounded byCγ ζ p+1W C1 γ−id C1C16and therefore we have the following estimate for(17)Cγ ζ p+1C1W C1 γ−id C1.We also observe that(18)can be bounded easily as followsTW{(∂x)γζp(∂x)γζ−∂xζp∂xζ}dx≤ W ∞ T∂xζp∂xζ((∂xγ−1◦γ)2−1)dx≤Cγ W ∞ ζ p+1C1γ−id C1.Combining this last estimate with the estimate for(19)we obtain ∂γ(∂−1x)γ(W)−∂γ(∂−1x)id(W) C1≤Cγ W C1 ζ p+1C1γ−id C1 which implies continuity of the map(12).For the continuity of the map(14)it is suﬃcient to observe that∂x(f◦γ−1)◦γ−∂x f ∞= ∂x f(∂xγ−1◦γ−1) ∞≤Cγ f C1 γ−id C1.Hence the continuity ofγ→∂Fγ,ζfollows.Furthermore the continuity inζof∂Fγ,ζis easier to prove.We rewrite the C1norm that we want to estimate as∂γFγ,ζ1−∂γFγ,ζ2C1≤ (∂−1x)γ{((∂x)γζp1(∂x)γζ1−(∂x)γζp2(∂x)γζ2)(∂x)γW} ∞+Cγ W C1 (∂x)γζp1(∂x)γζ1−(∂x)γζp2(∂x)γζ2 ∞(20)+ T W{(∂x)γζp1(∂x)γζ1−(∂x)γζp2(∂x)γζ2}By the deﬁnition of∂−1x theﬁrst term on the right hand side of this equality is bounded byCγ W C1 ∂xζp1∂xζ1−∂xζp2∂xζ2 ∞which,by adding and subtracting the appropriate terms can be estimated byCγ,ζ1,ζ2W C1 ζ1−ζ2 C1where Cγ,ζ1,ζ2depends only on C1norms ofγ,γ−1,ζ1andζ2.Similarly we estimate the second summand in(20)byCγ,ζ1,ζ2W C1 ζ1−ζ2 C1 and the third summand byCγ,ζ1,ζ2W ∞ ζ1−ζ2 C1.7Hence∂γF(γ,ζ)is continuous inζ.The continuity of∂γF(γ,ζ)inζcan be shown analogously.Therefore(ζp,F(γ,ζ))deﬁnes a continuous diﬀerentiable map in a neighborhood of(id,0).Remark1.There are diﬀerent forms of the Hunter-Saxton equation considered in the literature.The equation introduced by J.K.Hunter and R.Saxton in [HuSa]is∂x(∂t u+u∂x u)=12∂x((∂x u)2).(22)Note that12 T|∂x u0|2dx is a conserved quantity.Then the local well posedness of the periodic Cauchy problem for(22)with initial datau(x,0)=u0(x)is equivalent to that of∂t u+u∂x u=12(∂−1x)γ(((∂x)γ(ζ+¯u))2)˙γ=ζ+¯usuch that u=ζ◦γ−1+¯u is a solution to the Cauchy problem for(23)by a simple modiﬁcation of theorem2.Therefore for any given¯u(t)∈C1([0,∞),R),there is a unique solution u∈C0([0,T),C1(T))∩C1([0,T),L∞(T))to the Cauchy problem for the equation(22)such that T∂t u dx=¯u′for initial data u0∈C1(T).Remark2.In[Yi]Z.Yin shows that the initial value problem for(22)has a family of solutions in H s(T)for s>3/2.Our approach clariﬁes the uniqueness issue for this problem by identifying the relation between¯u′in(23)and a solution to the initial value problem for(22)(see remark1)as well as improving the well-posedness result from H s for s>3/2to C1.Remark3.The modiﬁed Hunter-Saxton equation∂x(∂t u+u p∂x u)=12(∂−1x)γ(((∂x)γ(ζ+¯u)p(∂x)γζ)˙γ=(ζ+¯u)p.82Well-posedness in H sIn proving theorem1we implement the same approach of rewriting the Cauchy problem for(mHS)as an initial value problem for an ordinary diﬀerential equa-tion on the product space D s×H s(T)where D s is the space of H s class diﬀeo-morphisms on T.Here we repeatedly use the Schauder ring property,Sobolev lemma and the following well-known property of Sobolev spaces that we refer to as the composition lemma.Lemma1.[Composition lemma]Let s>3/2,u∈H s andγbe a H s class bijection from T to T whose inverseγ−1is also of class H s.Then,u◦γ∈H s and the estimateu◦γ H s≤Cζ(1+ γ s H s) u H s(24) holds where Cγdepends only on C1norms ofγandγ−1.The next proposition is the H s version of proposition1.It is a restatement of the Cauchy problem for(mHS)with H s initial data as an initial value problem for an ordinary equation on D s×H s(T).Weﬁrst state the Cauchy problem for (mHS)on the product space D s×H s(T).Proposition2.A function u∈H s is a solution to the Cauchy problem(∂t u+u p∂x u)=p2(∂−1x ((ζ◦γ−1)p−1(∂x(ζ◦γ−1))2))◦γ.(25)with initial dataζ(x,0)=u0(x)andγ(x,0)=id x.Proof.Given a function u∈H s(T)which is a solution to the problem(25) there is aγ∈D s satisfying˙γ=u p◦γ,γ(x,0)=id x(see[BB]).We setζ=u◦γ.Then we have˙ζ=(∂tu+u p∂x u)◦γ.But u is a solution to the Cauchy problem for(mHS)therefore(25)holds for (γ,ζ).Conversely assume that we have a solution(γ,ζ)∈D s×H s(T)to(25) and let u=ζ◦γ−1.Thenζ=u◦γand therefore˙ζ=(∂tu+(˙γ◦γ−1)∂x u)◦γ.But since(γ,ζ)is a solution to(25)u is a solution to the Cauchy problem for (mHS).The next lemma is used in the proof of theorem1to show thatγ→(∂−1x)γis a continuous map from D s into L(H s−1,H s).9Lemma2.Let s>3/2.Then we have∂−1x(u◦γ−1)◦γ−∂−1x u H s≤C u H s−1 γ−id x H s( γ H s+1)for u∈H s−1(T)andγ∈D s.ing the identity in(15)we can write the H s norm to be estimated as follows:∂−1x(u◦γ−1)◦γ−∂−1x u H s= ∂−1x(u(∂xγ−1)) H s.Therefore by Poincar´e inequality,since∂−1x f has zero mean we have∂−1x(u∂xγ−u) H s≤ u(∂xγ−1) H s−1.(26) Finally using the Schauder ring property and Sobolev lemma with(26)we obtain ∂−1x(u◦γ−1)◦γ−∂−1x u H s≤C u H s−1 γ−id x H s.Now we are ready to proceed to the proof of theorem1.We will show that F(γ,ζ)is a bounded map with bounded linear directional derivatives which are continuous.Then theorem1follows from the fundamental theorem for ordinary diﬀerential equations on Banach spaces.Proof of theorem1.We start by showing that1F(γ,ζ)=(∂−1x)γ((∂x)γζp(∂x)γζ(∂x)γW)21+(∂−1x)γ{p(∂x)γ(ζp−1W)(∂x)γζ+(∂x)γζp(∂x)γW}(29)210deﬁne bounded linear maps.We estimate theﬁrst summand on the right hand side of∂γF(γ,ζ)(W) H s≃ (∂−1x)γ((∂x)γζp(∂x)γζ(∂x)γW) H s+ T W(∂x)γζp(∂x)γζdx (30) and ∂ζF(γ,ζ)W H s using the Schauder ring property,Poincar´e lemma and com-position lemma byCγ W H s ζ p(1+ ζ H s)H sas we estimated(27)above.For the absolute value term in(30)we apply Sobolev lemma to obtain the bound.Cγ W H s−1 ζ p+1H sHence the directional derivatives∂γF(γ,ζ)and∂ζF(γ,ζ)deﬁne bounded linear maps.The continuity of the directional derivatives can be proved as follows.We observe that(like in the previous section)to prove the continuity of γ→∂F(γ,ζ)it suﬃces to prove the continuity of the mapsγ→∂γ(∂−1x)γ∈L(H s,L(H s−1,H s))(31)γ→(∂−1x)γ∈L(H s−1,H s)(32)γ→(∂x)γ∈L(H s,H s−1)(33) in a neighborhood of(id,0)∈D s×H s(T).The identity in(15),Poincar´e inequality and Schauder ring property lead to the following estimate for the map in(32)(∂−1x)γf−∂−1x f H s≤Cγ f H s−1 γ−id H sand hence to the continuity of this map.In order to show that the map in(31)is continuous it is suﬃcient to obtain an appropriate estimate on the norm∂γ(∂−1x)γW−∂γ(∂−1x)id W H s≤ (∂−1x)γ{(∂x)γζp(∂x)γζ(∂x)γW}−∂−1x(∂xζp∂xζ∂x W) H s+ T W{(∂x)γζp(∂x)γζ−∂xζp∂xζ}dx .The second term on the right hand side above is bounded using Sobolev lemma and the Schauder ring property byγ−id H s.Cγ W H s−1 ζ p+1H sAdding and subtracting the appropriate terms and using Poincar´e inequality we obtain the following estimate for theﬁrst term(∂−1x)γ{(∂x)γζp(∂x)γζ(∂x)γW}−∂−1x(∂xζp∂xζ∂x W) H s≤ (∂−1x)γ((∂x)γζp(∂x)γζ(∂x)γW)−(∂−1x)γ(∂xζp∂xζ∂x W) H s(34) + (∂−1x)γ(∂xζp∂xζ∂x W)−∂−1x(∂xζp∂xζ∂x W) H s.(35)11For(35)lemma2and the Schauder ring property give us the following estimateW H s γ−id H s.Cγ ζ p+1H sApplying composition lemma and Poincar´e inequality to(34)we obtain (∂−1x)γ((∂x)γζp(∂x)γζ(∂x)γW)−(∂−1x)γ(∂xζp∂xζ∂x W) H sW H s γ−id H s.≤Cγ ζ p+1H sTherefore the map in(31)is continuous.It is not diﬃcult to obtain the following inequality which implies that the map in(33)is continuous using Schauder ring property(∂x)γW−∂x W H s−1≤Cγ W H s γ−id H s.In order to conclude that F(γ,ζ)deﬁnes a continuously diﬀerentiable map D s×H s into H s it is suﬃcient to observe that both directional derivatives ∂γF(γ,ζ)and∂ζF(γ,ζ)are continuous inζas well.This completes the proof of theorem1.3Analytic regularityIn this section we give a proof of theorem3that states the analytic regularity (i.e.,existence and uniqueness of analytic solutions for analytic initial data)of the Cauchy problem for(mHS).The classical Cauchy-Kowalevski theorem does not apply to the modiﬁed Hunter-Saxton equation(mHS).However a contraction argument on a scale of Banach spaces can be used for the nonlocal form(mHS)of this equation. The conditions under which such a contraction argument can be applied to prove an analytic regularity result are given in[Nis]and[BG]in the form of an abstract Cauchy-Kowalevski theorem.The following scale of Banach spaces is appropriate for our problem.Deﬁnition1.Let s be a positive real number.The collection{X s}s>0of Ba-nach spaces X s is called a decreasing scale of Banach spaces if s′<s implies X s⊂X s′and|||·|||s′≤|||·|||s.For s>0,let the spaces E s be deﬁned asE s= u∈C∞(T): T u dx=0and|||u|||s=sup k≥0 ∂k x.u H s s kLemma3.Let0<s<1.There is a constant c>0which is independent of s such that we have|||uv|||s≤c|||u|||s|||v|||sfor any u,v∈E s.We do not repeat the proof of this lemma here but refer to[HM1]where A.Himonas and G.Misio l ek prove the analytic regularity of solutions to the Cauchy problem for the Camassa-Holm equation.First we rewrite the equation(mHS)in a more convenient form.Let the operators P1and P2be deﬁned asP1(u):=−∂x u,P2(u):=∂−1x uand let u1:=u,u2:=∂x u.Then we can write the equation(mHS)in terms of (u1,u2)as follows∂t u1=F1(u1,u2):=12P2(u p−11u22)∂t u2=F2(u1,u2):=P1(u p1u2)+ps−s′|||u|||sholds.Lemma5.For any0<s<1,the estimate|||P2(u)|||s≤|||u|||sholds.Now we are ready to prove Theorem3.We will show that all three conditions of the abstract version of the Cauchy-Kowalevski theorem that we include in the appendix for the convenience of the reader as it is stated in[Nis],hold for (mHS)on the scale{X s}0<s<1.Proof of Theorem3.Let u=(u1,u2)and F=(F1,F2)in(36)and let X s be a decreasing scale of Banach spaces deﬁned as X s=E s×E s.Then we only need to verify theﬁrst two conditions of the abstract Cauchy-Kowalevski theorem since the map F(u1,u2)does not depend on t explicitly.Clearly,t−→F(t,u(t))=(F1(u1,u2),F2(u1,u2))is holomorphic if t−→u1(t)and t−→u2(t)are both holomorphic.Therefore,to verify theﬁrst con-dition of the abstract theorem,we only need to show that for s′<s,F1(u1,u2) and F2(u1,u2)are in E s′if u1,u2∈E s.We begin with estimates on F1:|||F1(u1,u2)|||s′=|||12P2(u p−11(u2)2)|||s′13≤c p (|||P 1(u p +11)|||s ′+|||P 2(u p −11(u 2)2)|||s ′).By Lemma 4and Lemma 5we have the following bound on this last term≤c p 1s −s ′|||u 1|||p +1s +c p |||u 1|||p −1s ′|||u 2|||2s ′.Similarly,for F 2,we have|||F 2(u 1,u 2)|||s ′=|||P 1(u p 1u 2)+p2|||u p −11u 22|||s ′≤c 12|||u 1|||p−1s ′|||u 2|||2s ′≤c 2|||u 1|||p −1s ′|||u 2|||2s ′.We proceed to establish the second condition of the abstract Cauchy-Kowalevski theorem.First we show that for some c independent of t the estimates|||F 1(u 1,u 2)−F 1(v 1,v 2)|||s ′≤cs −s ′|||u −v |||X s (38)hold.In order o obtain the estimate (37)we use the triangle inequality and Lemma 4with Lemma 5:|||F 1(u 1,u 2)−F 1(v 1,v 2)|||s ′≤c p |||P 1(u p +11)+P 2(u p −11u 22)−P 1(v p +11)−P 2(v p −11v 22))|||s ′≤c p (|||P 1(u p +11−v p +11)|||s ′+|||P 2(u p −11u 22−v p −11v 22)|||s ′)≤c ps −s ′(|||(u p +11−v p +11)|||s +|||u p −11(u 22−v 22)+(v p −11−u p −11)v 22|||s ′)≤c pand the assumption that|||u|||s<R and|||v|||s<R we have |||F1(u1,u2)−F1(v1,v2)|||s′≤c ps−s′|||u−v|||X swhere u=(u1,u2)and v=(v1,v2).The estimate(38)weﬁrst use triangle inequality|||F2(u1,u2)−F2(v1,v2)|||s′=|||P1(u p1u2)+p2v p−11v22|||s′≤c p(|||P1(u p1u2−v p1v2)|||s′+|||u p−11u22−v p−11v22|||s′).Then using Lemma4we estimate this last term by=c ps−s′(|||u p1(u2−v2)+(u p1−v p1)v2|||s+|||u p−11(u22−v22)+(u p−11−v p−11)v22|||s′)≤c ps−s′(|||u p1|||s|||u2−v2|||s+|||u p1−v p1|||s|||v2|||s+|||u p−11|||s′|||u22−v22|||s′+|||u p−11−v p−11|||s′|||v22|||s′).Again assuming that|||u|||s<R and|||v|||s<R we obtain|||F2(u1,u2)−F2(v1,v2)|||s′≤c p,Rform of the abstract theorem presented by F.Treves[Tre]has been improved by L.Nirenberg[Nir]and Nirenberg’s version has been further simpliﬁed by T.Nishida[Nis].We will use the following version of the abstract Cauchy-Kowalevski theorem from[Nis].Theorem4.Consider the Cauchy problem∂t u=F(t,u(t))u(0)=0.(39)Let{X s}0<s<1be a scale of decreasing Banach spaces,so that for any s′<s we have X s⊂X s′and|||.|||s′≤|||.|||s.Let T,R and C be positive numbers and suppose that F satisﬁes the following conditions1.)If for0<s′<s<1the function t−→u(t)is holomorphic in|t|<T and continuous on|t|≤T with values in X s andsup|t|≤T|||u(t)|||s<R,then t−→F(t,u(t))is a holomorphic function on|t|<T with values in X s′.2.)For any0<s′<s≤1and any u,v∈X s with|||u|||s<R,|||v|||s<R,sup |t|≤T |||F(t,u)−F(t,v)|||s′≤C1−s.Then there exists a T0∈(0,T)and a unique function u(t),which for every s∈(0,1)is holomorphic in|t|<(1−s)T0with values in X s,and is a solution to the initial value problem(39).References[AK]V.I.Arnold and B.A.Khesin,Topological Methods in Hydrodynamics, Springer Verlag,New York1998.[Arn]V.Arnold,Sur la g´e om´e trie diﬀ´e rentielle des groupes de Lie de dimen-sion inﬁnie et ses applications`a l’hydrodynamique desﬂuides parfaits, Ann.Inst.Fourier,Grenoble16(1966).[BB]J.P.Bourguignon and H.Brezis,Remarks on the Euler equation,J.Functional Analysis15(1974).[BG]M.S.Baouendi and C.Goulaouic,Sharp estimates for analytic pseudod-iﬀerential operators and application to Cauchy problems,J.Diﬀ.Eq.48 (1983).16[BSS1]R.Beals,D.Sattinger and J.Szmigielski,Acoustic scattering and the extended Korteweg-de Vries equation,,Advances in Mathematics,140 (1998).[BSS2]R.Beals,D.H.Sattinger and J.Szmigielski,Inverse scattering solutions of the Hunter-Saxton equation,Appl.Anal.78(2001).[Di]J.Dieudonn´e,Foundations of Modern Analysis,Academic Press,New York and London,1960.[Eb1] D.G.Ebin,Espace des m´e triques riemanniennes et mouvement desﬂu-ides via les vari´e t´e s d’applications,Centre de Math´e matiques de l’Ecole Polytechnique et Universit´e Paris81972.[Ee]J.Eells,A setting for global analysis,Bull.Amer.Math.Soc.72(1966).[EMa] D.G.Ebin and J.Marsden,Groups of diﬀeomorphisms and the motion of an incompressibleﬂuid,Ann.Math.92(1970).[HM1] A.A.Himonas and G.Misio l ek,Remarks on an integrable evolution equation,Geometry and analysis onﬁnite and inﬁnite dimensional Lie groups,Banach Center Publ.55,Polish Acad.Sci.,Warsaw(2002).[HM2] A.A.Himonas and G.Misio l ek,The Cauchy problem for an integrable shallow-water equation,Diﬀerential and Integral Equations14(2001).[HuSa]J.K.Hunter and R.Saxton,Dynamics of directorﬁelds,SIAM J.Appl.Math.51,No.6(1991).[HZ1]J.K.Hunter and Y.Zheng,On a completely integrable nonlinear hyper-bolic variational equation,Physica D79,1994.[K]T.Kato,Liapunov functions and monotonicity in the Navier-Stokes equation,Lecture Notes in Math.,1450,Springer,Berlin1990. [KaM]T.Kato and K.Masuda,Nonlinear evolution equations and analyticity I,Ann.de l’Inst.H.Poincar´e3(1986).[KM] B.A.Khesin and G.Misio l ek,Euler equations on homogeneous spaces and Virasoro orbits,Adv.Math.176,2003.[Mis1]G.Misio l ek,Classical solutions of the periodic Camassa-Holm equation, GAFA12(2002).[Mis3]G.Misio l ek,A shallow water equation as a geodesicﬂow on the Bott-Virasoro group,Journal of Geometry and Physics24(1996).[Nir]L.Nirenberg,An abstract form of the nonlinear Cauchy-Kowalevski the-orem,J.Diﬀerential Geometry6(1972).[Nis]T.Nishida,A note on a theorem of Nirenberg,J.Diﬀerential Geometry 12(1977).17[OK]V.Ovsienko and B.Khesin,Korteweg-de Vries superequations as an Euler equation,Functional Anal.Appl.21(1987).[Ovs1]L.V.Ovsjannikov,A singular operator in a scale of Banach spaces,Dokl.Akad.Nauk SSSR,163(1965).[Ovs2]L.V.Ovsjannikov,A nonlinear Cauchy problem in a scale of Banach spaces,Dokl.Akad.Nauk SSSR,200(1971);Soviet Math.Dokl.12 (1971).[R] E.G.Reyes,The soliton content of the Camassa-Holm and Hunter-Saxton equations,Proc.Inst.Math.NAS****43(2002).[Sat] D.H.Sattinger,Scaling,mathematical modeling,&Integrable systems, DMV-Seminar Series,Band28,Birkhauser1998.[Tay1]M.E.Taylor,Pseudodiﬀerential Operators and Nonlinear PDE, Birkh¨a user Boston1991.[Tay2]M.E.Taylor,Finite and inﬁnite dimensional Lie groups and evolution equations,Classroom Notes,University of North Carolina-Chapel Hill, Spring2003.[Tre] F.Treves,An abstract nonlinear Cauchy-Kovalevska theorem,Trans.Amer.Math.Soc.150(1970).[Tru] E.Trubovitz,The inverse problem for periodic potentials,Comm.Pure Appl.Math.30(1977).[Ya]T.Yamanaka,Note on Kowalevskaja’s system of partial diﬀerential equations,Comment.Math.Univ.St.Paul.9(1960).[Yi]Z.Yin,On the structure of solutions to the periodic Hunter-Saxton equa-tion,SIAM J.Math.Anal.36(2004).Feride TiglayUniversity of New Orleans,Department of Mathematics,Lakefront,New Orleans LA7014818。

2023年6月英语六级真题及参考答案

2023年6月英语六级真题及参考答案六级试卷采用多题多卷形式，大家核对答案时，找准具体选项内容，忽略套数！网络综合版：听力第一套Conversation One M: Hi Lily, how's the new apartment? W: It's okay. M:What? How can it be just okaywhen last week you were thrilledabout the place and keptnosting photos of it online? W:Well,【1】last week whenfmoved in, the apartment seemed cozy, justthe right size forone person. But nowit just seems tiny, shabby and solitary. M: Al that's the problem. You missyour roommates from university,don't you? W: I'm going to sound like G idiot【2】because Iused to complain to youall the time about how crowded ourdormitory room was, and about allthe things they did to irritate me, likewatching movies late at night withoutheadphones, or talking loudly early inthe morning. But now Imiss themterribly. M: Of course you do. That's perfectlynormal. When I got my first place,Iremember thinking I could ti wait tolive by myself and get away from myjuvenile roommates and all their annoyipghabits.【3】But then began issing them and feelinglonely and thinking that our dormitory was like paradise. Even though there were six of us guys inone small room. W: I thought it was just m who reltlike thiat. M: Look, you lived at home with us.And then you had three roommates.And this is your first time living alone.So i hard But your first apartmentis a milestone in your life. And youshould celebrate it. Tell me about theapartment. W: Actuaily, it's not bad. In fact, it'spretty adorable. Now that I have decorated it and it has et rjthing Ineed. I have a kitchen to cook in thebathroom al! to myself.And then anothes room with my bed at one endand the sofa, a small table and chairsat the C herend. M: That does sound adorable, and【4】Ican't wait to see it. And neither canmom and dad. Question 1: What was the woman'sfirst impression of the apartment? Question 2: Why does the womansay she's going to sound like an idiot? Question 3: What do we learn about the man when he left thedormitory to live on his own? Question 4: What is the man say hecan't wait to do. W: Welcome to our program book talk. Q5.ourgyest today is FrankJonesiditicbf our education system and the author of new book,How to reform our universitres. M: Hello, Susan. W: Frank, you support radicallychanging universities in America. Yes. Q6.I believe that the purpose of highereduad grefo prepare young peopleto enter the workforce and that ourcurrent system fails to do this, We'reallocating too many resou disciplines that don't match the needsof employers. W: I think your attitude to education isa bit cynical Frank. Surely the purpose of university is to prepareyoung people to participate fully incivic life rather than just to find wellpaid jobs. M: Susan, many young graduates struggle to find any job let alone agood one. The job markc isgrim.Particulaniior students who studythe arts. I agree that it isn't easy foryoungr gegple to find work, but youpropose closing down alt departments that aren't directly related to science and technology. Isthat really the solution? M: You're overstating my paint. Q7.My argument is that we need it use moer of our budget on areas like science and engineering. To do that, we needto take money from subjects likeliterature and music W: Q8,But the arts have value. They'rean important part of our culture.studying literature or music or sculpture might not result in a job inthatae But it helps young people tothink about the world in a.deeperway, which makes them b citizens and makes fora better society. M:l agree that the arts are valuable tosociety, but it's naive to think that notonly tk miost talented, but allstudents should study them at university level. The odds are verycompetitive, and most graduates willend up with a great deal of debt,obtaining a degree that has littlevalue on the job market. Question 5. What do we learn from theconversation about the man? Question 6. What does the manbelieve is the problem with the current AmericanSystem of Higher Education? Question 7. How should the educationbudget be allocated according to theman? Question 8. What does the woman saythe arts can do? Passage One Do you ever have the annoying feetingthat you don't have time to really thinkanymore? You're not alone.【Q9】A variet dtdrs have conspired to robus of time for reflectionourselves and our lives. preoccupied minds are rarely Silent.The average person receives hundredsof texts and voice messages a day. Andholidays for many of us are action-packed weeks more likely full of familyactivities than opportunities fortranquility and contemplation.【Q10】Regular reflection,howe,underlies all great professionals. It's a prerequisite for you to recharge yourmental batteries. See things in a newlight and tap into your creativity. Almost all of the great advisors that Ihave studied have found ways to getaway from it all and contemplate theirlife and work. Some researchers in thefield of creativity, in fact, believe thatinsight occurs during the reflection and relaxation that follows aCeriod of intense actvity. Schedule your time for reflection aboutyourwork ora particular proiect you're engaged in. I usually biock outhalf an hour. Don't answer the phone.Push your papers to the side. Sketch,make lists, draw mind maps of ideasthat come to you. At the end, write down any emerging ideas. When you're alone, stop worrying andthink. A lot of our downtime is spentworryingabout troublesc ne thihgs inour lives or fantasizing aboat how we'dlike our lives to be.【Q11】 Revisitthings during moments of relaxationafter a periodnof intense work. This iswhenwe are the most creative. Question 9 What do we learn about thefeeling that one doesn't heeitime tothink anymore? Questica 10 What trait do all greatprofessionals share? Question 11 What is some researchersbelieve is conducive to creative ideas? Passage Two had post offices The first opened in 1859 in asettlement founded by migrants searching for gold,Life could be unpredictable outwest. Gold failed to appear. Drought ruinedfarmers, and settlers clashed with_NativeAmericans. On the settlement's location now stands asprawling University campus. Amid all thechanges, one feature remained constant: thepostal service. The maps tracing America'swestward expansion are telling in 1864 therewere few postat branches on land controlledby Native Americans, which still accountedfor most of the West. Over the next 25 years,post offices grew quickly. Colonization'of theWest could be regarded as a result of biggovernment rather than pioneers.【13】Asfederal subsidies and land grants temptedpeople into the deserts and plains, the postkept them connected. In the mid-19th century, the Post OfficeDepartment was far from a centralizedbureaucracy. To keep up with migrationpatterns, postal services were added toexisting businesses. 【14】The federal government commissioned private wagons themail. Short term contracts were granted tolocal businessman to act as postmasters.These partnerships enabled the mail to quickly followmigrants helping knit togetherremote parts of the country. Mr. Bellavance, a digital historian, wrote abook on the history of the US postal service. 【15】 He used the data science to analyzehistorical trends, Most strikingly he built anaccompanying website, complekw;Tinteractive maps.They show readers-howwithin a generation the postal service helpedcolonize a continent. These online interactivemaps illustrate the formative power of snailmail. Q12 What does the passage say AboutColorado before it became a state? Q13 How did the postal service contribute toAmerica's westward expansion? Q14 What did the federal government:do tomeet the increasing demand for the postalservice in the West? Q15 What did Mr. Bellavance do to study thehistory of the US postal service? 听力演讲1 In last week's lecture, we discussed reasons whypeople forget things. This week we will discuss asurprising reason why we might remember somethings, anxiety. Think about something as simple asbuying a coffee. That may not seem like an experience that would make a deep impression onyour memory. But anxiety could change that. Q16.In fact, a new study suggests that people withhigher anxiety levels mightremembertertain information better than people with lower anxietylevels. That's because higher levels of anxiety may makepeople moresusceptible to negative feelings, putting them in a more negative state of mind. Thatin turn, may make them able to better remembersome events. Let's take a closer look at that newstudy now. Q17. In this study, tseardhersstarted by giving 80 undergraduate students ananxiety test. The test measure the participantsanxiety levels over the proceeding two weeks. Then, to test memory, the participants were showna series of neutral words one at a time. Some of thewords were printed onto photos of negative scenes,meaning images that could affect their emotionsnegatively, such as a photo of a car accident, or acemetery. The rest of the words were printed ontophotos of neutral scenes, such as a photo of a lakeor trees. Neutral words included words like table ordesk that don't elicit emotion. Later, the participants were asked to think back tothe words they were shown earlier, which causedthem to reenter either a negative or neutral mindset. The participants were then presented withanother set of neutral words, and their memory ofthese new words was tested. The researchers found that the new words presented to people in a negative mindset werebetter remembered by people with higher levels ofanxiety than those with lower levels of anxiety. In other words, when highly anxious individualstook in otherwise emotionallyneutral informationthat was presented to them, it became colored bytheir negative mindset, making them remember theinformation better. But these same effects were notseen in people with low levels of anxiety. Q18. Previous studies havefound that extremeevels of anxiety such as those experienced bypeople diagnosed with an anxiety disorder can bequite detrimentalto memory and cognitive performan But the highly anxious people in thisstudy represent individuals who are managing theiranxiety and for whom anxiety is not. a seriousproblem. Question 16. What does the speaker say the newstudy suggests? Question 17. What did researchers do first in thenew study? Question 18. What do we learn from previousstudies aboutlanxiety? Over the past 20 years, the u ternet hasgradually become a dominant featureof our lives. It has changed how wecommunicate with each other. And ithas definitely transformed the way wedo business with each other:Marketinghas also changed in a number of ways. For instance, in the past, consumershad to call a phone number and patiently wait on hold in order to getthe information they wanted. [Q19]Today, they want the informationimmediately. They'll go to the company's sociaLmediapdc nifostcomments and questions expecting toreceive an immediate response. If theydon't get their questions answeredsoon they'll move on to anothercompany that will answer themquickly. Marketing departments today need tofollow technological development.Forexample, this year smartphone issmarter than last years. s fariving cars are now on the road. Marketershave to do research on which techncingies:are coming into bsing,otherwise, they risk being leit behindin the virtual dust. Marketing has also changed due to theimportance of video. People don't justwant to read text. They walt to watchthings happening. Companies now have to explore how they can use videoon a consistent basis to share information about their sinesses.Fortunately, it's extremely easy to shoot something these days. All youneed is a smartphone. But what's the result of all this? Shorteraitention spans? We aren't the samepeople that we were 20 yedi ago. Notonly have we grown accustomed togetting the information we want instantaneously, our attention spansare much shorter. If something doesn'tcaptulc ourattention within a few seconds. We're on to the next piece ofcontent.[Q20]Marketers need to figureout ways to speak directly to the customer's emotions and they need tofigure out how to do that as quickly aspossible. Once people are emotionallyengaged, they'll stick with you. If marketingi has changed this much inthe past 20 years, imagine what thenext 20 years will bring li ai recentsurvey, only 9% of marketers could saywith confidence that their marketingefforts were actually working. Theirconfidence is being shaken becausethe rules of the game change everyyear. That's why [Q21]it'simportant for marketers to pay attention to the latest technological devel and consider collaborating with technological innovators. That way,they'll be moving at the samepace asthe tech industry. Question 19 What does the speaker sayabout today's consumers? Question 20 How do marketers captureconsumers'attention as quickly aspossible? Question 21 What does the speakersuggest marketers do to meet futurechalletes? 演讲3 You might be surprised to learn that [Q22] thebenefits offriendships extend beyond people's sociallifeand into their work, which is interestingwhen cd lili the extent to which people sacrifice friendships, or at least the time they spendwith friends because of the exte edihairsthey'redevoting to work. Just last week, rwas remarking toa colleague that I'm content with only one socialengagement per week. But according to recentresearch, that's evidently not enough. In an initial study of more than 700 respondents,scholars from an American university [Q23] analyzethe imrf thst:fiends as opposed to family haveon sel dem Jahd well-being. Friends came outsubstantially on top. That's because to be someone's mate is a voluntary act. Unlike familywho people rarely get to choose. The researchersfound that when people choose to cultivate andmaintain supportive friendships with an individual,it means that the person is valued and worthy oftheir limited time. Such sentiments of value andworthiness boost our self-esteem. The second study comprised more than 300 participants. It proved that the better we feel aboutourselves, the more likely we will perform our jobconfidently andcompetently. This follow-up studyfound that [Q24] non-work friends even improvedpeople's job satisfaction. They have as much of animpact on how much they love their jobs, as do thefriends they have at work, despite not actuallybeing at our place of work. These types of friendstend to be our preferred outlet fo nni aboutwork-related mattersyThis is an avenue that maynot be available at the office. So even though friendships can be easy to neglectwhen confronted-by pressures at work, or evenpressures at home, neglecting our friends can turnout to be harmful and counterproductive. That'swhy when determining how to create a better work-life balance, we need to consider not only how tobalance work and family demands, but also how tocultivate and sustain supportive friendships. It's for employees for flexible work arrangements. It's irrelevant whether their need for a desired scheduleis due to say, parenting responsibilities, or a craving to hang out with their best mate. Whatmatters is the opportunity to engage in a nourishingactiyity outside of work. That will definitely have afollow-on effect at work. Q 22 What does the speaker say is interesting? Q 23 What did researchers from an Americanuniversity analyze in their initial study? Q 24 What did the second study find aboutmon-work friends? Q 25 What does the speaker suggest managers do? 参考答案： 1.A) She is drawn to its integration of design andengineering. 2.D) Through hard work. 3.C) It is long-lasting. 4.A) Computer science. 5.B) He is well known to the public. 6.D) Serve as a personatassistant. 7.D) He has little previous work experience. 8.C) He has a high proficiency in several languages. 9.A) They have fewer rules and pressures. 10.D) They deprive kids of the opportunity todevelop team spirit. 11.C) Let them participate in some less risky outdooractivities. 12.B) Tech firms intentionally design products tohave short lifespans. 13.C) List a repairability score of their product. 14.D) Take the initintive to reduce electronic waste. 15.A) It can be solved. 16.B) How to prevent employees from cyberloafing. 17.C) Cyberloafing may relieve employees of stress. 18.A) Taking mini-breaks means better jobperfontance. 19.D) There were no trees. 20.B) He founded a newspaper and used it topromote his ideas. 21.B) The state government declared it the officialArbor Day. 22.B)They moved ou of Africa about 60,000 yearsago. 23.D) The discovery of two modern human teeth inChina. 24.A) There must have been some reason for humanmigration.25.D) What path modern humans took to migrateout of America.听力第二套参考答案：作文：　心理健康 Mental well-being is regarded as a state of health where a person is able to address normal stresses in daily life. Recently,this state has been grasped as much attention as physical health. Obviously, there are several factors that affect people's mental well-being. Firstly, a strong contributor to mental well-being refers to the state of a person'susual environ-ment. Adverse environmental circumstances can lea negative effectson psychological wellness. Living in a positive social environment, in contrast, can provide protection against mental challenges. Secondly, people's lifestyle can also impact their mental health. Smoking, a poor diet, alcohol con-sumption, substance use, and risky sexual behavior may result in psychological harm. Smoking, a poor diet, alcohol consump-tion, substance use, and risky sexual behavior may result in psychological harm. Worse, such behaviors have been linked to depression. In conclusion, because mental health is so important to general wellness, it's important that you take care of your mental health. Talking therapy, meditation and maintaining a positive outlook on life all contribute to people mental health. With a positive mental state, all areas of life will go towards active de-velopment. 友好的讨论 When faced with differing opinions, we should try to reach agreement through friendly discussion and reasonable argu-ment. In our daily life, it is common to see college students struggling with a polite and logical way when their views differ from others'. Apparently, this issue has sparked public con-cerns. Friendly discussion allows individuals to share their perspec-tives and opinions in a respectful manner. This can lead to a better understanding of each other's viewpoints and poten-tially even finding common ground. In addition, reasonable ar-gument allows individuals to present evidence and logic to sup-port their position, which can help persuade others to see their point of view. However, it is important to note that not all disagreements can be resolved through discussionand argu-ment alone. In some cases, compromise may be necessary to reach a resolution that satisfies all parties involved. To sum up, friendly discussion and reasonable argument, to a large extent, are of great use. We should be open-minded and engaged in such practices. 教育的目标 Education has played an increasingly crucial role in modern so-ciety. We aim education on different levels at cultivating the to-be successors of our global village. One important goal that education is trying to achieve is help students master the ways to acquire knowledge. Of all the capabilities one can develop to acquire knowledge in being educated, three sorts are of the greatest significance.First of all, students who are receiving education definitely know that they are always ignorant of some branches in th eocean of knowledge, which can keep them modest and more willing to explore their unfamiliar realms, even deeper if they've already done so. Moreover, students can imitate what their teachers or professors do in or our of class and then gradually acquire the ability to undertake more scientific re-search and intellectual inquiries alone. Last but not least,youngsters who are accustomed to being educated at school or college are more likely to keep studying as a life-long habit,which will have a substantially positive effect on their own life and the future of the human world. In my perspective, education is one of the most marvelous social inventionsthat ever existed in human history. Without it, the whole globe can never continue developing further in a civilized and prosperous direction.星火英语版：听力部分（共2套）第一套1.B) It was warm and comfortable.2.B) She misses her roommates she used to complain about.3.C) He had a similar feeling to the woman's.4.A) Go to see the woman's apartment.5.D) He has published a book recently.6.C) It has not prepared young people for the job market.7.A) More of the budget should go to science and technology.8.D) Cultivate better citizens.9. A) It is quite common.10. B) Engaging in regular contemplation.11. D) Reflecting during ones relaxation.12. C) There existed post offices.13. D) It kept people in the deserts and plains connected.14. B) It commissioned private wagons to carry the mail.15. C) He examined its historical trends with data science.16. A) Higher levels of anxiety may improve people's memory.17）C) They measured the participants' anxiety levels.18.B) Extreme levels of anxiety can adversely affect cognitive performance.19. D) They expect to get instantaneous responses to their inquiry.20. C) Speaking directly to their emotions.21. B) Keep up with the latest technological developments.22. D) Friendships benefit work.23. A) The impact of friends on people's self-esteem.24. D) They increase people's job satisfaction.25. A) Allow employees to have a flexible work schedule.第二套1. A) She is drawn to its integration of design and engineering.2.D) Through hard work.3.C) It is long-lasting.4.A) Computer science.5.B) He is well known to the public.6.D) Serve as a personal assistant.7.D) He has little previous work experience.8.C) He has a high proficiency in several languages.9.A) They have fewer rules and pressures.10.D) They deprive kids of the opportunity to develop team spirit.11 C) Let them participate in some less risky outdoor activities.12. B) Tech firms intentionally design products to have short lifespans.13. C) List a repairability score of their product.14. D) Take the initintive to reduce electronic waste.15. A) It can be solved.16.B) How to prevent employees from cyberloafing.17.C) Cyberloafing may relieve employees of stress.18. A) Taking mini-breaks means better job perfontance.19.D) There were no trees.20.B) He founded a newspaper and used it to promote his ideas.21.B) The state government declared it the official Arbor Day.22.B）They moved ou of Africa about 60,000 years ago.23.D) The discovery of two modern human teeth in China.24.A) There must have been some reason for human migration.25.D) What path modern humans took to migrate out of America.翻译部分（共3套）1.中国文化出口近年来，越来越多的中国文化产品走向全球市场，日益受到海外消费者的青睐。

2017年专业英语八级考试真题及答案

2017年专业英语八级考试真题及答案PART I LISTENING COMPREHENSIONSECTION A MINI-LECTUREIn this section you will hear a mini-lecture. You will hear the mini-lecture ONCE ONLY. While listening to the mini-lecture, please complete the gap-filling task on ANSWER SHEET ONE and write NO MORE THAN THREE WORDS for each gap. Make sure the word(s) you fill in is (are) both grammatically and semantically acceptable. You may use the blank sheet for note-taking.You have THIRTY seconds to preview the gap-filling task.Now listen to the mini-lecture. When it is over, you will be given THREE minutes to check your work.SECTION B INTERVIEWIn this section you will hear TWO interviews. At the end of each interview, five questions will be asked about what was said. Both the interviews and the questions will be spoken ONCE ONLY. After each question there will be a ten-second pause. During the pause, you should read the four choices of A, B, C and D, and mark the best answer to each question on ANSWER SHEET TWO.You have THIRTY seconds to preview the choices.Now, listen to the first interview. Questions 1 to 5 are based on the first interview.1. A. Comprehensive. B. Disheartening. C. Encouraging. D. Optimistic.2. A. 200. B. 70. C. 10. D. 500.3. A. Lack of international funding.B. Inadequate training of medical personnel.C. Ineffectiveness of treatment efforts.D. Insufficient operational efforts on the ground.4. A. They can start education programs for local people.B. They can open up more treatment units.C. They can provide proper treatment to patients.D. They can become professional.5. A. Provision of medical facilities.B. Assessment from international agencies.C. Ebola outpacing operational efforts.D. Effective treatment of Ebola.Now, listen to the second interview. Questions 6 to 10 are based on the second interview.6. A. Interpreting the changes from different sources.B. Analyzing changes from the Internet for customers.C. Using media information to inspire new ideas.D. Creating things from changes in behavior, media, etc.7. A. Knowing previous success stories.B. Being brave and willing to take a risk.C. Being sensitive to business data.D. Being aware of what is interesting.8. A. Having people take a risk.B. Aiming at a consumer leek.C. Using messages to do things.D. Focusing on data-based ideas.9. A. Looking for opportunities.B. Considering a starting point.C. Establishing the focal point.D. Examining the future carefully.10.A. A media agency.B. An Internet company.C. A venture capital firm.D. A behavioral study center.PART II READING COMPREHENSIONSECTION A MULTIPLE CHOICE QUESTIONSIn this section there are three passages followed by fourteen multiple choice questions. For each multiple choice question, there are four suggested answers marked A, B, C and D. Choose the one that you think is the best answer and mark your answers on ANSWER SHEET TWO.PASSAGE ONE(1) It’s 7 pm on a balmy Saturday night in June, and I have just ordered my first beer in I Cervejaria, a restaurant in Zambujeira do Mar, one of the prettiest villages on Portugal’s south-west coast. The place is empty, but this doesn’t surprise me at all. I have spent two weeks in this area, driving along empty roads, playing with my son on empty beaches, and staying in B&Bs where we are the only guests.(2) No doubt the restaurant, run by two brothers for the past 28 years, is buzzingin July and August, when Portuguese holidaymakers descend on the Alentejo coast. But for the other 10 months of the year, the trickle of diners who come to feast on fantastically fresh seafood reflects the general pace of life in the Alentejo: sleepy, bordering on comatose.(3) One of the poorest, least-developed, least-populated regions in western Europe, the Alentejo has been dubbed both the Provence and the Tuscany of Portugal. Neither is accurate. Its scenery is not as pretty and, apart from in the capital Evora, its food isn’t as sophisticated. The charms of this land of wheat fields, cork oak forests, wildflower meadows and tiny white-washed villages, are more subtle than in France or Italy’s poster regions.(4) To travel here is to step back in time 40 or 50 years. Life rolls along ata treacly pace; there’s an unnerving stillness to the landscape. But that stillness ends abruptly at the Atlantic Ocean, where there is drama in spades. Protected by the South West Alentejo and Costa Vicentina national park, the 100 km of coastline from Porto Covo in the Alentejo to Burgau in the Algarve is the most stunning in Europe. And yet few people seem to know about it. Walkers come to admire the views from the Fisherman’s Way, surfers to ride the best waves in Europe, but day after day we had spectacular beaches to ourselves.(5) The lack of awareness is partly a matter of accessibility (these beaches are a good two hours’ drive from either Faro or Lisbon airports) and partly to do with a lack of beachside accommodation. There are some gorgeous, independent guesthouses in this area, but they are hidden in valleys or at the end of dirt tracks.(6) Our base was a beautiful 600-acre estate of uncultivated land covered in rock-rose, eucalyptus and wild flowers 13km inland from Zambujeira. Our one-bedroom home, Azenha, was once home to the miller who tended the now-restored watermill next to it. A kilometre away from the main house, pool and restaurant, it is gloriously isolated.(7) Stepping out of the house in the morning to greet our neighbours – wild horses on one side, donkeys on the other – with nothing but birdsong filling the air, I felt a sense of adventure you normally only get with wild camping.(8) “When people first arrive, they feel a little anxious wondering what they are going to do the whole time,” Sarah Gredley, the English owner of estate, told me. “But it doesn’t usually take them long to realise that the whole point of being here is to slow down, to enjoy nature.”(9) We followed her advice, walking down to the stream in search of terrapins and otters, or through clusters of cork oak trees. On some days, we tramped uphill to the windmill, now a romantic house for two, for panoramic views across the estate and beyond.(10) When we ventured out, we were always drawn back to the coast – the gentle sands and shallow bay of Farol beach. At the end of the day, we would head,sandy-footed, to the nearest restaurant, knowing that at every one there would be a cabinet full of fresh seafood to choose from – bass, salmon, lobster, prawns, crabs, goose barnacles, clams … We never ate the same thing twice.(11) A kilometre or so from I Cervejaria, on Zambujeira’s idyllic natural harbour is O Sacas, originally built to feed the fishermen but now popular with everyone. After scarfing platefuls of seafood on the terrace, we wandered down to the harbour where two fishermen, in wetsuits, were setting out by boat across the clear turquoise water to collect goose barnacles. Other than them, the place was deserted – just another empty beauty spot where I wondered for the hundredth time that week how this pristine stretch of coast has remained so undiscovered.11.The first part of Para. 4 refers to the fact that ______.A.life there is quiet and slowB.the place is little knownC.the place is least populatedD.there are stunning views12.“The lack of awareness” in Para. 5 refers to ______.A.different holidaying preferencesB.difficulty of finding accommodationC.little knowledge of the beauty of the beachD.long distance from the airports13.The author uses “gloriously” in Para. 6 to ______.A.describe the scenery outside the houseB.show appreciation of the surroundingsC.contrast greenery with isolationD.praise the region’s unique feature14.The sentence “We never ate the same thing twice” in Para. 10 reflects the ______of the seafood there.A.freshnessB.delicacyC.tasteD.variety15.Which of the following themes is repeated in both Paras. 1 and 11?A.Publicity.ndscape.C.Seafood.D.Accommodation.PASSAGE TWO(1) I can still remember the faces when I suggested a method of dealing with what most teachers of English considered one of their pet horrors, extended reading. The room was full of tired teachers, and many were quite cynical about the offer to work together to create a new and dynamic approach to the place of stories in the classroom.(2) They had seen promises come and go and mere words weren't going to convince them, which was a shame as it was mere words that we were principally dealing with. Most teachers were unimpressed by the extended reading challenge from the Ministry, and their lack of enthusiasm for the rather dry list of suggested tales was passed on to their students and everyone was pleased when that part of the syllabus was over. It was simply a box ticking exercise. We needed to do something more. We neededa very different approach.(3) That was ten years ago. Now we have a different approach, and it works. Here’s how it happened (or, like most good stories, here are the main parts. You have to fill in some of yourself employing that underused classroom device, the imagination.) We started with three main precepts:(4) First, it is important to realize that all of us are storytellers, tellers of tales. We all have our own narratives – the real stories such as what happened to us this morning or last night, and the ones we have been told by others and we haven’t experienced personally. We could say that our entire lives are constructed as narratives. As a result we all understand and instinctively feel narrative structure. Binary opposites – for example, the tension created between good and bad together with the resolution of that tension through the intervention of time, resourcefulness and virtue – is a concept understood by even the youngest children. Professor Kieran Egan, in his seminal book ‘Teaching as Storytelling’ warns us not to ignore this innate skill, for it is a remarkable tool for learning.(5) We need to understand that writing and reading are two sides of the same coin: an author has not completed the task if the book is not read: the creative circle is not complete without the reader, who will supply their own creative input to the process. Samuel Johnson said: A writer only begins a book. A reader finishes it. In teaching terms, we often forget that reading itself can be a creative process, just as writing is, and we too often relegate it to a means of data collection. We frequently forget to make that distinction when presenting narratives or poetry, and often ask comprehension questions which relate to factual information – who said what and when, rather than speculating on ‘why’, for example, or examining the context of the action.(6) The third part of the reasoning that we adopted relates to the need to engage the students as readers in their own right, not as simply as language learners; learning the language is part of the process, not the reason for reading. What they read must become theirs and have its own special and secret life in their heads, a place where teachers can only go if invited.(7) We quickly found that one of the most important ways of making all the foregoing happen was to engage the creative talents of the class before they read a word of the text. The pre-reading activities become the most important part of the teaching process; the actual reading part can almost be seen as the cream on the cake, and the principle aim of pre-reading activities is to get students to want to read the text. We developed a series of activities which uses clues or fragments from the text yet to be read, and which rely on the student’s innate knowledge of narrative, so that they can to build their own stories before they read the key text. They have enough information to generate ideas but not so much that it becomes simply an exercise in guided writing; releasing a free imagination is the objective.(8) Moving from pre-reading to reading, we may introduce textual intervention activities. ‘Textual Intervention’ is a term used by Rob Pope to describe the process of questioning a text not simply as a guide to comprehension but as a way of exploring the context of the story at any one time, and examining points at which the narrative presents choices, points of divergence, or narrative crossroads. We don’t do this for all texts, however, as the shorter ones do not seem to gain much from this process and it simply breaks up the reading pleasure.(9) Follow-up activities are needed, at the least, to round off the activity, to bring some sense of closure but they also offer an opportunity to link the reading experience more directly to the requirements of the syllabus. Indeed, the story may have been chosen in the first place because the context supports one of the themes that teachers are required to examine as part of the syllabus – for example, ‘families’, ‘science and technology’, ‘communications’, ‘the environment’and all the other familiar themes. There are very few stories that can’t be explored without some part of the syllabus being supported. For many teachers this is an essential requirement if they are to engage in such extensive reading at all.(10) The whole process – pre-, while and post reading – could be just an hour’s activity, or it could last for more than one lesson. When we are designing the materials for exploring stories clearly it is isn’t possible for us to know how much time any teacher will have available, which is why we construct the activities into a series of independent units which we call kits. They are called kits because we expect teachers to build their own lessons out of the materials we provide, which implies that large amounts may be discarded. What we do ask, though, is that the pre-reading activities be included, if nothing else. That is essential for the process to engage the student as a creative reader..(11) One of the purposes of encouraging a creative reading approach in the language classroom is to do with the dynamics we perceive in the classroom. Strategictheorists tell us of the social trinity, whereby three elements are required to achieve a dynamic in any social situation. In the language classroom these might be seen as consisting of the student, the teacher and the language. Certainly from the perspective of the student – and usually from the perspective of the teacher – the relationship is an unequal one, with the language being perceived as placed closer to the teacher than the student. This will result in less dynamic between language and student than between language and teacher. However, if we replace ‘language’ with narrative and especially if that is approached as a creative process that draws the student in so that they feel they ‘own’ the relationship with the text, then this will shift the dynamic in the classroom so that the student, who has now become a reader, is much closer to the language – or narrative – than previously. This creates a much more effective dynamic of learning. However, some teachers feel threatened by this apparent loss of overall control and mastery. Indeed, the whole business of open ended creativity and a lack of boxes to tick for the correct answer is quite unsettling territory for some to find themselves in.16.It can be inferred from Paras. 1 and 2 that teachers used to ______.A.oppose strongly the teaching of extended readingB.be confused over how to teach extended readingC.be against adopting new methods of teachingD.teach extended reading in a perfunctory way17.The sentence “we all understand and instinctively feel narrative structure”in Para. 4 indicates that ______.A.we are good at telling storiesB.we all like telling storiesC.we are born story-tellersD.we all like listening to stories18.Samuel Johnson regards the relationship between a writer and a reader as ______(Para. 5).A.independentB.collaborativeC.contradictoryD.reciprocal19.In Para. 7, the author sees “pre-reading” as the most important part of readingbecause _____.A.it encourages students’ imaginationB.it lays a good foundation for readingC.it can attract students’ attentionD.it provides clues to the text to be read20.“Textual Intervention” suggested by Rob Pope (in Para. 8) is expected tofulfill all the following functions EXCEPT ______.A.exploring the contextB.interpreting ambiguitiesC.stretching the imaginationD.examining the structurePASSAGE THREE(1) Once again, seething, residual anger has burst forth in an American city. And the riots that overtook Los Angeles were a reminder of what knowledgeable observers have been saying for a quarter century: America will continue paying a high price in civil and ethnic unrest unless the nation commits itself to programs that help the urban poor lead productive and respectable lives.(2) Once again, a proven program is worth pondering: national service.(3) Somewhat akin to the military training that generations of American males received in the armed forces, a 1990s version would prepare thousands of unemployable and undereducated young adults for quality lives in our increasingly global and technology-driven economy. National service opportunities would be available to any who needed it and, make no mistake, the problems are now so structural, to intractable, that any solution will require massive federal intervention.(4) In his much quoted book, “The Truly Disadvantaged,” sociologist William Julius Wilson wrote that “only a major program of economic reform” will prevent the riot-prone urban underclass from being permanently locked out of American economic life. Today, we simply have no choice. The enemy within and among our separate ethnic selves is as daunting as any foreign foe.(5) Families who are rent apart by welfare dependency, job discrimination and intense feelings of alienation have produces minority teenagers with very little self-discipline and little faith that good grades and the American work ethic will pay off. A military-like environment for them with practical domestic objectives could produce startling results.(6) Military service has been the most successful career training program we’ve ever known, and American children born in the years since the all-volunteer Army was instituted make up a large proportion of this targeted group. But this opportunity may disappear forever if too many of our military bases are summarily closed and converted or sold to the private sector. The facilities, manpower, traditions, and capacity are already in place.(7) Don’t dismantle it: rechannel it.(8) Discipline is a cornerstone of any responsible citizen’s life. I was taught it by my father, who was a policeman. May of the rioters have never had any at all. As an athlete and former Army officer, I know that discipline can be learned. More importantly, it must be learned or it doesn’t take hold.(9) A precedent for this approach was the Civilian Conservation Corps that worked so well during the Great Depression. My father enlisted in the CCC as a young man with an elementary school education and he learned invaluable skills that served him well throughout his life. The key was that a job was waiting for him when he finished. The certainty of that first entry-level position is essential if severely alienated young minority men and women are to keep the faith.(10) We all know these are difficult times for the public sector, but here’s the chance to add energetic and able manpower to America’s workforce. They could be prepared for the world of work or college – an offer similar to that made to returning GI after Word War II. It would be a chance for 16- to 21-year-olds to live among other cultures, religions, races and in different geographical areas. And these young people could be taught to rally around common goals and friendships that evolve out of pride in one’s squad, platoon, company, battalion – or commander.(11) We saw such images during the Persian Gulf War and during the NACC Final Four basketball games. In military life and competitive sports, this camaraderie doesn’t just happen; it is taught and learned in an atmosphere of discipline and earned mutual respect for each other’s capabilities.(12) A national service program would also help overcome two damaging perceptions held by America’s disaffected youth: the society just doesn’t care about minority youngsters and that one’s personal best efforts will not be rewarded in our discriminatory job market. Harvard professor Robert Reich’s research has shown that urban social ills are so pervasive that the upper 20 percent of Americans – the “fortunate fifth” as he calls them – have decided quietly to “secede”from the bottom four-fifths and the lowest fifth in particular. We cannot accept such estrangement on a permanent basis. And what better way to answer skeptics from any group than by certifying the technical skills of graduates from a national service training program?(13) Now, we must act decisively to forestall future urban unrest. Republicans must put aside their aversion to funding programs aimed at certain cultural groups. Democrats must forget labels and recognize that a geographically isolated subgroup of Americans – their children in particular – need systematic and substantive assistance for at least another 20 years.(14) The ethnic taproots of minority Americans are deeply buried in a soil of faith and loyalty to traditional values. With its emphasis on discipline, teamwork, conflict resolution, personal responsibility and marketable skills development, national service can provide both the training and that vital first job that will reconnect these Americans to the rest of us. Let’s do it before the fire next time.21.According to the author, “national service” is comparable to “militarytraining” because they both cultivate youngsters’ ______.A.good gradesB.self disciplineC.mutual trustD.work ethic22.The author cites the example of his father in order to show ______.A.the importance of disciplineB.the importance of educationC.the necessity of having strong faithD.the effectiveness of the program23.According to the author, a national service program can bring the followingbenefits to America’s youngsters EXCEPT ______.A.increase in incomeB. a sense of responsibilityC.confidence and hopeD.practical work skills24.According to the context, what does “the fire” refer to (Para. 14)?A.Discrimination.B.Anger.C.Riots.D.Aversion.SECTION B SHORT ANSWER QUESTIONSIn this section there are eight short answer questions based on the passages in SECTION A. Answer each question in NO MORE THAN TEN WORDS in the space provided on ANSWER SHEET TWO.PASSAGE ONE25.What does Para. 2 tell us about the restaurant business on the Alentejo coastthroughout the year?26.According to Para. 5, what are the two main reasons of the Alentejo’sinaccessibility?PASSAGE TWO27.What does “It was simply a box ticking exercise” mean in Para. 2?28.Paras. 4-6 propose three main precepts for the now approach. Please use ONEphrase to summarize each of the three precepts.29.What does the author suggest to shift the dynamic in the classroom (Para. 11)?PASSAGE THREE30.What is the purpose of the program proposed by the author (Paras. 1-3)?31.What does the word “it” in “Don’t dismantle it: rechannel it.” refer to(Para. 7)?32.What do Robert Reich’s findings imply (Para. 12)?PART III LANGUAGE USAGEThe passage contains TEN errors. Each indicated line contains a maximum of ONE error. In each case, only ONE word is involved. You should proof-read the passage and correct it in the following way:For a wrong word, underline the wrong word and write the correct one inthe blank provided at the end of the line.For a missing word, mark the position of the missing word with a “∧” signand write the word you believe to be missing in theblank provided at the end of the line.For an unnecessary word, cross the unnecessary word with a slash “/” and put the word in the blank provided at the end of the line.ExampleWhen∧art museum wants a new exhibit, (1) an__________it never buys things in finished form and hangs (2) never__________them on the wall. When a natural history museumwants an exhibition, it must often build it. (3) exhibit__________Proofread the given passage on ANSWER SHEET THREE as instructed.PART IV TRANSLATIONTranslate the underlined part of the following text from Chinese into English. Write your translation on ANSWER SHEET THREE.我小的时候特别盼望过年，往往是一过了腊月，就开始掰着指头数日子。

遇到的困难如何解决和结果英语范文

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遇到困难和解决方法英语作文

遇到困难和解决方法英语作文Challenges and Solutions.Life is an endless journey filled with challenges and obstacles. It is how we face and overcome thesedifficulties that shape our character and determine our success. Throughout history, countless individuals have encountered adversity, but it is those who persevered and found innovative solutions who have left an indelible mark on the world.Identifying Challenges.The first step in overcoming challenges is toaccurately identify their nature. This requires self-awareness and the ability to analyze situations objectively. Is the challenge a lack of resources, a conflict with another person, or a personal limitation? Once the natureof the challenge is understood, we can begin to explore potential solutions.Creative Problem-Solving.Solving challenges often requires creative thinking and unconventional approaches. Instead of relying ontraditional methods, it is essential to explore new perspectives and consider different angles. This may involve seeking advice from others, researching innovative solutions, or experimenting with different ideas. By approaching challenges with an open mind, we increase the likelihood of finding effective solutions.Collaboration and Support.No one is immune to challenges, and seeking help from others can be invaluable in finding solutions. Collaborate with friends, family, colleagues, or mentors who may have faced similar obstacles. Share your ideas and listen to their perspectives. Sometimes, a fresh perspective can provide the necessary insight to overcome a challenge.Overcoming Personal Limitations.Personal limitations can often be the most challenging obstacles to overcome. They may include fears, insecurities, or self-doubt. To overcome these limitations, it isessential to practice self-reflection and challengenegative thoughts. Identify the root causes of your limitations and develop strategies to overcome them. This may involve setting small, achievable goals, seeking professional help, or practicing positive affirmations.Persistence and Resilience.Finding solutions to challenges is rarely a linear process. There will be setbacks and disappointments along the way. It is during these times that persistence and resilience are crucial. Do not give up easily. Learn from your mistakes and use them as opportunities for growth. The ability to bounce back from setbacks and maintain apositive attitude is essential for overcoming even the most difficult challenges.Learning from Successes and Failures.Every experience, whether a success or a failure, provides valuable lessons. Analyze what worked well and what could have been done better. Identify patterns and insights that can be applied to future challenges. By learning from our mistakes, we can avoid repeating them and improve our problem-solving abilities.Conclusion.Overcoming challenges is an integral part of human existence. By embracing challenges, seeking innovative solutions, collaborating with others, overcoming personal limitations, and maintaining persistence and resilience, we can emerge from adversity stronger and more capable. The solutions we find to the challenges we encounter not only shape our present but also lay the foundation for ourfuture success. Remember, it is not the absence of challenges but the ability to overcome them that defines our true potential.。

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《The Problems of Inequality》高考优秀英语作文The Problems of InequalityIn the modern society, inequality exists everywhere and it is a very serious issue. Even though most people know that inequality is wrong and should be abolished, there are still many cases of inequality happening in our everyday lives. Thus, tackling the problem of inequality should be dealt with earnestly.First of all, laws should be passed to protect people from discrimination from different aspects such as gender, race, and social or economic status. People should not be judged by their identity or background but on their own merits. The government should also allocate more resources to vulnerable groups such as disabled or marginalized individuals. This would provide them with the opportunity to improve their living standard and contribute back to the society.Besides, educational institutions should also take proactive steps to foster equality among students so as to create a fair learning environment. Educational resources should be made available to every student regardless of their social status, and educational institutions should encourage inclusive practices and respect for diversity.Moreover, it is important to raise public awareness on the issue of inequality. Public campaigns can be organized to promote understanding and acceptance of equality. We should work together to encourage compassion and respect for all.In conclusion, inequality can only be eliminated if everyone takes responsibility for it and takes actions. With ev eryone’s help, we can create a society where everyone can live and pursue their dreams with equal opportunities.。

Mathematical Tools for Applied Multivariate Analysis chapter 6

As again recalled from Chapter 1, the multiple regression equation
Yi = bo +
baii+b.Xt2
is a linear equation for predicting values of Y that minimize the sum of the squared errors
6.2
THE MULTIPLE REGRESSION PROBLEM
We are now ready to work through the details of the sample problem in Chapter 1 dealing with the relationship of employee absenteeism Y, to attitude toward the firm Xi and number of years employed by the firm X2. To simplify our discussion, the basic data, first shown in Table 1.2, are reproduced in Table 6.1. As recalled from the discussion in Chapter 1, here we are interested in 1. finding a regression equation for estimating values of the criterion variable Y from a linear function of the predictor variables Xi and X2 ; 2. determining the strength of the overall relationship; 3. testing the significance of the overall relationship; 4. determining the relative importance of the two predictors Xi and X2 in accounting for variation in Y. 6.2.1 The Estimating Equation

小学上册第十四次英语第二单元真题(含答案)

小学上册英语第二单元真题(含答案)考试时间：90分钟（总分:110）A卷一、综合题(共计100题共100分)1. 听力题：A reaction that releases energy is called an ______ reaction.2. 听力题：My brother is very ________.3. 听力题：The phase change from solid to liquid is called ______.4. 选择题：Which animal is known as the king of the jungle?A. LionB. TigerC. ElephantD. Bear5. 听力题：In a chemical equation, the substances that react are called ______.6. 选择题：What do we call the area around the equator?A. TropicsB. PolesC. ZonesD. Continents答案:A7. 选择题：What do we call the scientific study of plants?A. BotanyB. ZoologyC. EcologyD. Geography答案: A8. 填空题：I find ________ (社会学) very interesting.9. 选择题：What is the name of the small, winged insect that produces honey?A. AntB. FlyC. BeeD. Wasp答案:C10. 听力题：The baby is ________ in the crib.11. 选择题：What is 50 ÷ 10?A. 4B. 5C. 6D. 7答案:B12. 填空题：He is a _____ (评论员) on a popular podcast.13. 听力题：The ______ teaches us about international relations.14. 填空题：The coach, ______ (教练), encourages us to do our best.15. 选择题：What is the name of the fictional superhero from Gotham City?A. SupermanB. SpidermanC. BatmanD. Ironman16. 听力题：A ______ is a type of animal that has a pouch.Many plants have adapted to survive in ______ climates. (许多植物已适应在极端气候中生存。

小学下册第十四次英语第一单元期中试卷

小学下册英语第一单元期中试卷英语试题一、综合题(本题有50小题，每小题1分，共100分.每小题不选、错误，均不给分)1 What is the primary color of grass?A. YellowB. GreenC. BlueD. Red答案:B2 The __________ is a famous coastal city.3 What do you call a person who writes stories?A. NovelistB. PoetC. AuthorD. All of the above4 What is 2 + 2?A. 3B. 4C. 5D. 6答案: B5 The __________ (历史的探究) broadens horizons.6 My ________ (玩具) is easy to carry around.7 The sunflowers are ______ (tall) and bright.8 I enjoy painting my ________ (玩具类型).9 What do bees collect from flowers?A. NectarB. PollenC. HoneyD. Water答案: A10 The _____ (飞盘) is great for playing outside.11 I enjoy playing ______ (棋盘游戏) with my family during the weekends.12 The __________ was a period of exploration and colonization. (大航海时代)13 The ancient Romans are known for their contributions to ________ (法律).14 The __________ (历史的传承) enriches culture.15 Matter can exist in three states: solid, liquid, and ______.16 The _______ (青蛙) is often found near water sources.17 He is reading a _____ (书).18 The skunk can spray a _______ (臭味).19 What do you call a tool used to measure length?a. Scaleb. Rulerc. Stopwatchd. Compass答案:B20 What do we call a person who repairs cars?A. DoctorB. MechanicC. PilotD. Chef21 The candy is very _______ (sticky).22 The __________ is a famous city known for its beaches and nightlife. (迈阿密)23 A __________ solution has more solute than it can dissolve.24 What is the name of the famous mountain range in South America?A. AndesB. RockiesC. HimalayasD. Alps答案：A25 The chemical symbol for neon is _____.26 What is the main ingredient in pancakes?A. FlourB. SugarC. EggsD. Milk答案:A27 What is the capital city of Mongolia?A. UlaanbaatarB. ErdenetD. Choibalsan28 What do you call a person who sells goods?A. MerchantB. BuyerC. CustomerD. Trader答案: A29 The chemical symbol for ruthenium is ______.30 The invention of the microscope opened doors in _____.31 I see _____ (星星) at night.32 What is a common pet that purrs?A. DogB. RabbitC. CatD. Hamster答案: C33 I love ________ (learning) new things.34 The ________ blooms in spring.35 The ________ loves to play.36 A battery stores ______ energy.37 The sun is very ___ (bright) today.38 The weather is ______ today. (nice)39 What do we call the practice of keeping bees?A. BeekeepingC. Honey farmingD. Pollination40 A cheetah is the fastest _______ on land, capable of incredible speed.41 They are going to ________ a movie.42 What do you call a piece of meat from a pig?A. BeefB. ChickenC. PorkD. Lamb答案:C43 I enjoy ___ (reading) before bed.44 What do we call the act of encouraging innovation?A. CreativityB. EntrepreneurshipC. InventionD. All of the Above答案：D45 Who is known for saying "I have a dream"?A. Nelson MandelaB. Martin Luther King Jr.C. Albert EinsteinD. Mahatma Gandhi46 The sun is _____ after the rain. (shining)47 We visit the ______ (博物馆) often.48 My aunt loves to __________. (陪伴)49 Rockets are used to launch spacecraft into ______.50 What is the opposite of 'light'?A. HeavyB. DarkC. BrightD. Clear51 What is a baby frog called?A. CubB. KittenC. TadpoleD. Puppy52 My brother is an excellent __________ (学生).53 The __________ is where most of the world's earthquakes occur. (环太平洋火山带)54 A chemical property describes how a substance ______.55 The ________ (公共设施) support community needs.56 I have a _____ (question/answer) for you.57 A _____ (植物展示) can highlight conservation efforts.58 What do you call a story that is told in verse?A. NovellaB. PoemC. ProseD. Narrative答案: B59 What is the name of the famous British author known for his fantasy novels?A. J.R.R. TolkienB. C.S. LewisC. J.K. RowlingD. All of the above答案:D60 I love reading books. My favorite book is about ________ (动物). It teaches me about ________ (自然).61 How many legs does a cat have?A. TwoB. FourC. SixD. Eight答案:B62 What is the main function of the heart?A. Pump bloodB. Digest foodC. Filter wasteD. Produce hormones答案: A63 trial Revolution began in _____. The Indu64 ________ (植物观察) is a fun hobby.65 The __________ (历史的反思) leads to growth.66 What is the main ingredient in bread?A. SugarB. FlourC. MilkD. Water答案:B67 The capital city of Eswatini is ________ (斯威士兰的首都城市是________).68 A _______ is a common houseplant.69 The _____ (小猫) pounces on a toy mouse.70 We like to _______ games at home.71 My _____ (老师) gives us homework every day.72 My sister has a strong interest in __________ (文学).73 What do you call a baby cat?A. PuppyB. KittenC. CubD. Fawn答案: B74 I enjoy visiting the ________ (文化中心) for classes.75 Which animal is known for its speed?A. TortoiseB. HareC. SlothD. Snail答案:B76 The process of fermentation converts sugar into ______.77 The _____ (绿色) leaves of the plants are very vibrant.78 The __________ is a long area of land that is surrounded by water on three sides.79 The candy is ___ (sour).80 The process of evaporation involves heat and ______.81 A compound that can donate hydroxide ions is called a ______.82 My uncle is a __________ (历史学家).83 What do you call a large, flightless bird?A. EagleB. OstrichC. SparrowD. Penguin84 What is the name of the famous American author known for "Little Women"?A. Louisa May AlcottB. Jane AustenC. Charlotte BrontëD. Emily Dickinson答案：A85 The Pacific Ocean is the largest _______ on Earth.86 She loves to _______ (travel) overseas.87 What is the capital of Brazil?A. Rio de JaneiroB. BrasíliaC. São PauloD. Salvador88 What do we call a scientist who studies the earth and its composition?A. BiologistB. GeologistC. ChemistD. Physicist答案:B89 What do you call a person who repairs pipes?A. ElectricianB. MechanicC. PlumberD. Builder答案:C90 What is the name of the fairy tale character who lost her glass slipper?A. Snow WhiteB. CinderellaC. Little Red Riding HoodD. Rapunzel91 The chemical symbol for rhenium is _____.92 What is the name of the process by which plants lose water?A. TranspirationB. PhotosynthesisC. RespirationD. Germination答案:A93 What is the capital of South Africa?A. Cape TownB. PretoriaC. JohannesburgD. Durban答案:B94 The ancient Greeks built _______ to discuss politics. (集会所)95 We have a ______ (丰富的) series of events planned.96 Certain plants can be used to create natural ______ against pests. (某些植物可以用来创建自然屏障，抵御害虫。

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arXiv:math/0512243v1 [math.CA] 12 Dec 2005
R. Fuchs’ problem of the Painlev´e equations from the ﬁrst to the ﬁfth
Yousuke Ohyama
Graduate School of Information Science and Technology, Osaka University
R. Fuchs’ Problem Let y(t) be an algebraic solution y(t) of a Painlev´e equation. Find a suitable transformation x = x(z, t) such that the corresponding linear diﬀerential equation
We show such rational transformations correspond to almost all of algebraic solutions of the Painlev´e equations from the ﬁrst to ﬁfth up to the B¨acklund transformations. The cases of the degenerate ﬁfth Painlev´e equation and the Laguerre type solution of the ﬁfth Painlev´e equation are exceptional. We need exponential type transformations since the monodromy data is decomposable.
1
example of such solutions is Picard’s solutions of the sixth Painlev´e equation [18]. R. Fuchs calculated the monodromy group of the linear equation corresponding to Picard’s solutions [4]. It seems that R. Fuchs’ paper [4], whose title is the same as the famous paper [3], has been forgotten for long years. Recently Mazzocco found again his result independently but his paper was not referred in her paper [15]. In [4], R. Fuchs proposed the following problem:
2
Goursat [7]. Goursat found many rational transformations which keep the hypergeometric equations. His list is incomplete and Vidu¯nas made a complete list of rational transformations [21].
Recently, Kitaev, jointing with Andreev, constructed many algebraic solutions of the sixth Painlev´e equation, which include known ones and new ones, by rational transformations of the hypergeometric equations [1], [14]. At least now, we do not know whether R. Fuchs’ problem is true or not for the sixth Painlev´e equation. We do not have negative example for R. Fuchs’ problem. Kitaev’s transformation is a generalization of classical work by
The parabolic cylinder equation (the Weber equation), the Bessel equation and the Airy equation are reduced to the conﬂuent hypergeometric equations by rational transformations of the independent variable (see (4), (11) and (12)). These classical formula can be considered as conﬂuent version of the Goursat transformations. Our result is an analogue of such classical formula in the Painlev´e analysis.
In this paper we will study a conﬂuent version of [1] and we show that R, Fuchs’ problem is true for the Painlev´e equation from the ﬁrst to the ﬁfth. We will classify all rational transformations which change the conﬂuent hypergeometric equations to linear equations of the Painlev´e type from the ﬁrst to the ﬁfth. Compared with the sixth Painlev´e equation, we obtain less transformations since the Poincar´e rank of irregular singularities of the linear equations of the Painlev´e type is up to three. We have up to sixth order rational functions which change the conﬂuent hypergeometric equations to linear equations of the Painlev´e type.
If R. Fuchs’ problem is true, algebraic solutions of the Painlev´e equations can be considered as a kind of a generalization of Schwarz’ solutions. Schwarz’ solutions are constructed by rational transformations which change hypergeometric equations to hypergeometric equations. Algebraic solutions of the Painlev´e equations are constructed by rational transformations which change hypergeometric equations to linear equations of the Painlev´e type and we can calculate the monodromy data of the linear equation explicitly.
Riemann calculated the monodromy group for the Euler-Gauss hypergeometric equation. Moreover, monodromy groups of the Euler-Gauss hypergeometric equation are the polyhedron groups when it has algebraic solutions [19]. Schwarz’ ﬁfteen algebraic solutions are obtained from simple hypergeometric equations by rational transformations of the independent variable. We will study an analogue of Schwarz’ solutions in the Painlev´e case.
d2v dz2
=
Q(t, y(t), z)v
is changed to the form without the deformation parameter t:
d2u dx2
=
Q˜(x)u.
Here v = dz/dx u. (See the lemma 5.3.)
Picard’s solution is algebraic if it corresponds to rational points of elliptic curves. For three, four and six divided points, the genus of algebraic Picard’s solutions is zero. R. Fuchs showed that for algebraic Picard’s solutions whose genus are zero the corresponding linear equations are reduced the EulerGauss hypergeometric equations by suitable rational transformations.