Perturbation Methods and First Order Partial Differential Equations

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Perturbation Methods and First Order Partial Di?erential Equations David Holcman ?Ivan Kupka ?November 26/2002Abstract In this paper,we give explicit estimates that insure the existence of solutions for ?rst order partial di?erential operators on compact manifolds,using a viscosity method.In the linear case,an explicit integral formula can be found,using the characteristics curves.The solution is given explicitly on the critical points and the limit cycles of the vector ?eld of the ?rst order term of the operator.In the nonlin-ear case,a generalization of the Weitzenb¨o ck formula provides pointwise estimates that insure the existence of a solution,but the uniqueness question is left open.Nevertheless we prove that uniqueness is stable under a C 1perturbation.Finally,we give some examples where the solution fails to exist globally,justifying the need to impose conditions that warrant global existence.The last result reveals that the zero order term in the ?rst order operator is necessary to obtain generically bounded solutions.

Contents

1Introduction

21.1Notations ....................................52The linear case:main theorem

72.1proof of existence and uniqueness .......................

102.2proof of regularity ...............................

102.3explicit formulas on the recurrent set (12)

3The nonlinear case13

3.1Existence of the solution for the elliptic PDE (14)

3.2Regularity estimates (17)

4Study of limit when the viscosity parameter converges to zero20 5Some examples23

5.1Ergodic?elds (25)

6When the zero order term vanishes25

6.1The gradient case with symmetries (29)

6.2Negative case (29)

7Conclusion29 8Appendix130 9Appendix230 1Introduction

In this paper,we consider a compact orientable Riemannian manifold(V,g),a smooth vector?eld b on V or a parametrized vector?eld b:R xV—>TV,the tangent bundle of V,and a smooth positive function c:V—>R or a parametrized positive smooth function c:R xV—>R.?g is the negative Laplacian:

?g=?div g grad g

where div g denotes the divergence operator with respect to the volume form associated to the metric g and grad g the gradient with respect to g.

We shall study the limit of the solutions u?asεtends to0through positive values of the linear equations:

??g u?+g+cu?=f on V(1) where f is a given continuous function on the manifold V or of the non linear equations:??g u?(x)+g+c(uε(x),x)uε(x)=0,x∈V(2)

Heuristically,the limits if they exists,”should”be solutions of the?rst order partial di?erential equation:

g+cu=f on V(3)

in the linear case and:

g+c(u(x),x)u(x)=0,x∈V(4) in the nonlinear case.The linear case is obviously a particular case of the non linear one.

Historically,Cauchy devised a powerful method to?nd local solutions of the Cauchy or initial value problem for?rst order partial di?erential equations(linear or non linear) using the characteristic curves which were the solutions of an ordinary di?erential system in the?rst jet bundle,called the characteristic system(see[3]).The initial data were given on a hypersurface in the bundle of the?rst jets of functions on the space of the partial di?erential equation and then propagated along the characteristic curves from the hypersurface.In the case of equation(4),using the natural coordinate system (x1,...,x m,u,p1,...,p m),on the jet space,the characteristic system for equation(4)can be expressed as follows:

dx n

dt =

m

k=1p k b k(u,x)

dp n

?x n (u,x)+p n

?b k

?u

(u,x)+c(u,x))?

?c

see equation(33)below.This integral equation can be solved using the Picard?xed point theorem and give us a Lipschitz continuous solutions under assumptions which are stronger than the ones needed to prove the existence using the viscosity method developed below.

The standard viscosity method is the approach using an elliptic partial di?erential equation to?nd solutions for a?rst order partial di?erential equation.Other methods such as the one mentioned in the last paragraph,exist but the advantage of the viscosity method is that,explicit geometrical conditions and explicit constants in inequalities are derived.Under these explicit conditions,the existence and local uniqueness of solutions are established.We prove that in certain types of?rst order linear partial di?erential equation,the solution is unique and a representation of this solution can be found using the trajectories of a vector?eld.

First order partial di?erential equations similar to equation4and3were considered in relation to the KAM theory by Kuksin in[16].Here V is a m-dimensional torus endowed with the?at metric b is a translation-invariant purely imaginary vector?eld.c must satis?es the condition(dx is the Haar measure)

V

c(x)dx >>max{osc(c),1}.(7)

Under some incommensurability assumptions on the coe?cients of the?eld in the standard basis of the lie algebra of the torus and the condition7,Kuksin proves that equation3 has a unique analytic solution for any analytic f.Moreover sup-norm estimates of the solution are given in terms of the sup-norm of f.The method of proof is based on the theory of similar equations on the circle S1.Because we are on a torus Fourier series can be used to represent the functions and this is how the estimates are derived.

Forni in[6]studies equation3with c=0on compact surfaces endowed with a Rieman-nian metric.He assumes that the vector?eld b preserves the volume measure associated to the metric and that its singularities are nondegenerate saddle points only.When f is su?ciently regular and its mean value vanishes,there exists a unique solution(up to a constant of course).

In the following paper,there are no restrictions on the underlying manifold V(except compactness),b is a general Morse-Smale vector?eld.Hence in particular it is not volume preserving.Also in our case c cannot be taken to be0.It has to satisfy condition min

V c>b0(de?ned in the notations below)which is more stringent than7.This condition means that c must be larger than the variation of b.In the case considered by Kuksin, this variation is0because b is self-parallel.

Some of the results exposed here were announced in[9].Let us?nally mention that when the function f in the right hand side of equation(2)is proportional to u,we get an eigenvalue problem.The analysis of this problem has been done in the general case in [10,11].

The paper is organized as follow.In the?rst section of the paper we study the limits asεgoes to0of the solutions of equation(1)to prove the existence and uniqueness of so-lutions for the linear equation(3).Then we prove the uniqueness of the solutions.In the

second section we take up the existence and uniqueness of solutions of the nonlinear el-liptic equation(2).The estimates needed to achieve this are provided by the Weitzenb¨o ck formula and its generalization to covariant tensors of valence two.Then as in the?rst section we examine the limits of solutions of equation(2)asεgoes to0and prove the ex-istence of solutions for the nonlinear?rst order equation(4).In the third and last section, we give various examples in the nonlinear case where a solution can exists in an open set but not globally.When the?eld is ergodic,we give a simple result about the behavior of the solution on the manifold.Finally in order to understand how relevant is the zero order term,we consider the case where it is zero.In that case,the viscosity method does not converge to any bounded solutions.Generically,a?rst order linear partial di?erential equation with no zero order term cannot have bounded solutions.

1.1Notations

Throughout this paper we shall use the following notations:

1)Metric and tensors:

d g=distanc

e function de?ned on V by the metric

||?||,g=norm and scalar product associated to g

vol g=volume measure associated to g

grad=gradient operator associated to g

div g=divergence operator associated to g

?=Levi-Civita covariant derivative associated to g

R=curvature tensor of

θ(b):=Lie derivative operator associated to the vector?eld b

θ(b)u=du(b)

For any coordinate system x1,....,x m:O—>R on V:

g=g ij dx i?dx j

g ij=inverse of the matrix g ij

vol g=2

?x j

?i=??

?x l

?i?j??j?i=R···l ijk·dx k??

If T is a p-covariant,q-contravariant vector ?eld:

T =T ?,...?j 1,..,j q i 1,..i p dx i 1?...?dx i p +1?e j 1?...?e j q

?T =?i 1T ?,...?j 1,..,j q

i 2,..i p +1dx i 1?...?dx i p +1?e j 1?...?e j q

where (e 1,...,e m )is the frame ?eld associated to the coordinates x 1,....,x m ,on O .

2)Norms:g induces a scalar product function and a norm function on any tensor bundle on V.Let t,τbe two tensors of the same type,at the same point of V.

g =scalar product of t and τ

||t ||g =norm of t

To any tensor ?eld T on V is associated the function:

x ∈V —>||T ||g ∈[0,+∞[.

||T ||∞=max x ∈V ||T (x)||g ||T ||L ∞=ess-sup x ∈V

||T (x)||g

||T ||L 2= V ||T (x)||2g vol g (dx ) 1

||T ||2L 2+||?T ||2L 2

3)Constants:

b 0=sup {g |X ∈T V,||X ||g =1}

β=||?b ?λ

||g c 0=inf λ∈R ,x ∈V

c (λ,x )>0f 0=||

d f ||∞+

||f ||∞||dc ||∞?λ||∞.||f ||∞

A function f:V—>R,will be called Lipschitz continuous if:

|f(x)?f(y)|

sup

x,y∈V

.

c(P)

6.If p:R→V n is a periodic trajectory of b with minimal period T,for all t∈R

u(p(t))=C(T)

C(s)

ds+

1

C(s)

ds,for all t,

(11)

where C(t)=exp[? t0c(p(s))ds].

Proof:First we will prove that any sub-sequences of u?contains a converging sub-sequence whose limit u in the weak L2topology,is a Lipschitz continuous function sat-isfying equation(9).In order to show this we need to establish some a priory estimates which follow easily from the maximum principle.The following is easy:

min f

V

min c

V

(12)

There exists a sub-sequence of u?which converges weakly to a function u and for all φ∈H1,

? V u??gφ? V div(φb)u?+ Vφcu?= Vφf

u satis?es for allφ∈H1,

? V div(φb)u+ Vφcu= Vφf

In these equations the integrations are with respect to the measure de?ned by the metric g on V.

Now we will prove that||du?||∞is also bounded.Let us note here that using a stronger lower bound on c,one could prove that||?du?||g is also bounded in L∞which would imply di?erentiability of the limit.To estimate||du?||2∞,we shall consider the maximum of the function||du?||2,following a standard method(see[17,13,19]).

Starting with the Weitzenb¨o ck formula:

1

2?g(||du

?||2g)+?||?du?||2g+?Ricci(du?,du?)+c||du?||2g+g

=?

1

Note that at a maximum point P of the function||du?||2g,?g(||du?||2g)(P)≥0and (θ(b)||du||2g)(P)=0.Hence we get using the relations(12):

||du?||∞=||du?(P)||g≤||d f||∞+||f||∞||dc||∞

c0??r0?b0(13) Now we drop the auxiliary assumption that f∈C1and prove that for any Lipschitz con-

tinuous f:

||du?||∞≤||d f||L∞+||f||∞||dc||∞

c0??r0?b0

We use the following lemma(probably well known)proved in the appendix:

Lemma1Let k:V—>R,be a Lipschitz continuous function with M as Lipschitz bound. Then for any neighborhood U of k in C0(V),anyε>0,there exists a C∞function h:V—>R,contained in U and admitting M+ε,as Lipschitz bound.

Hence with the lemma,we can?nd a sequence{f n|n∈N}of smooth functions on V,converging to f in C0(V),f n allowing||d f||L∞+1

c0

2

].From this it follows that uεis Lipschitz continuous and that:

||duε||∞≤||d f||L∞+||f||∞||dc||∞

c0??r0?b0 Because||u?||g,||du?||g are uniformly bounded in?forε∈[0,c0?b0

c0

2.1proof of existence and uniqueness

Now we prove that there is only one function u:V→R,which is a weak solution of equation(9)and is Lipschitz continuous.We have just shown that such u’s exist.Taking any one of them we will give an integral representation of the function u along the tra-jectories of b which will be used to compute the value of the function u at the stationary points and along the periodic trajectories of b.Since u is Lipschitz continuous,it is abso-lutely continuous along any C1curve and almost everywhere di?erentiable.If x:R→V is any trajectory of b,the function:t∈R→u(x(t))is absolutely continuous and satis?es the equation:

du(x(t))

Proof of the lemma:The?rst statement is easy and left to the reader.Let us prove3?rst.Denote by v0any tangent vector to V.Let v:R→T V,be the vector?eld:

v(t)=Tφt(v

)

We have:

?t v=?v b

d||v(t)||2g

dt≤2b0||v(t)||2g

From this it follows that if t≥0:

||v(t)||g≤||v(0)||g e b0t

Reversing the time,we get for t<0:

||v(t)||g≤||v(0)||g e b0|t|

This last inequality implies3).

Let us prove2).Let p:R→V be a periodic trajectory of b of(minimal)period T and let λbe an eigenvalue of the monodromy of p.There exists a tangent vector v0∈T p(0)V?C, v0=0,such that TφT(v0)=λv0and Tφ?T(v0)=1

|λ|≤e b0T

This two inequalities imply assertion2).

Now we resume the proof of assertion4).Let{φt|t∈R}denote the?ow of b.?(t,x)=?t(x).

For any x∈V:

u(x)= 0?∞K(t,x)dt(15) where K:R×V→R,is the function:

K(t,x)=f(φ(t,x))e? 0t c(φ(s,x))ds

Let K t(x)=K(t,x).K is a C∞function.Its di?erential in x is:

dK t(x)=e? 0t c(φ(s,x))ds[d f(φ(t,x))T xφt? 0

t

dc(φ(s,x))T xφs ds]

T x φt is the tangent mapping of the di?eomorphism ?t :V—>V,at x.The second integral is that of the curve:s ∈R →dc (φ(s,x))T x φs ∈T ?x V,the cotangent space of V at https://www.wendangku.net/doc/5b3805076.html,ing the statement 3)of the Lemma(2),for t ≤0:

||dK t (x )||≤e (c 0?b 0)t [||f ||1+|t |||c ||1]

This estimate,uniform for all x ∈X and all t ≤0,shows that u is continuously di?erentiable on V because of the assumption c 0>b 0and that:

du (x )= 0

?∞

dK (t,x )dt

2.3explicit formulas on the recurrent set

Now we can specialize the formula(15)to di?erent kinds of trajectories.Let P be a singular point of b.Then the curve x:R →V,x (t )=P for all t is a trajectory and

u (P )=u (x (0))= 0?∞f (x (s ))e ? 0s c (x (τ))dτ= 0?∞

f (P )e ? 0s c (P )dτ=

f (P )C (s )ds + t 0f (p (s ))C (s )

ds =+∞ n =0 ?nT ?(n +1)T f (p (s ))C (s )

ds =e ?(n +1)C (T ) T 0f (p (s ))C (s )ds =e ?C (T )

C (s )

ds u (p (t ))=C (t )[e ?C (T )

C (s )ds + t 0f (p (s ))1?e ?C (T ) T

t f (p (s ))1?e ?C (T ) t 0

f (p (s ))

Remark1:Let us de?ne the following real number:

B0=inf

g b0(g)=inf

g∈Riem

max{|g|X∈T V,|X|=1},

where Riem denotes the Riemannian structure on V.Lemma2shows that B0is strictly positive(bigger than the minimum of the logρand logρ

3.1Existence of the solution for the elliptic PDE

To prove the ?rst part of the theorem,about the existence of a solution to the ?rst order partial di?erential equation,we will built a sequence of solutions of some nonlinear elliptic partial di?erential equations with a small viscosity coe?cient and later on,we will let this small parameter converges to zero.

Now the assumptions (1-2)imply that there exists two strictly positive roots of the second order equation:βX 2?X (c 0?b 0?γ)+f 0=0.The same property will be true for the equation:βX 2?X (c 0?b 0?εr 0?γ)+f 0=0provided that 0≤ε≤ε=c 0?b 0?γ?2√r 0

(17)For ε∈[0,

ε[,

||du ||∞≤R (ε).

For any ε0in ]0,max c R xV ≤u ?≤max f

V

2

+||?du k ||2g +Ricc (du k ,du k )=g (18)

Then

?g (||du k ||2

g )2

θ(b k ?1)(||du k ||2g )+[g +du k ?b dλg ]|λ=u k ?1

(19)

where c k(x)=c(u k(x),x)and b k(x)=b(u k(x),x).In order to prove that the gradient of the

sequence is bounded,we evaluate the formula(19)at a maximum point P of||?u k||2g. Because?g(||u k||2g)(P)≥0andθ(b k)(||du k||2g)(P)=0,we get:

(c0??r0?b0?β||du k?1||∞)||du k||2∞≤||d f||∞+||f||∞?λ|.||du k?1||∞](20) Inequality(20)implies the following estimates for?small enough,for all k≥1,

(c0?b0?r0??β||du k?1||∞)||du k||∞≤f0+γ||du k?1||∞(21) Elementary properties of homographic recurrent sequences show that||du k?1||∞≤R(?)

implies||du k||∞≤R(?)for anyε∈]0,

c0?b0?εr0.

A simple computation proves that f0

2?g w2n+θ(b u

n

)w2n/2+c u

n

w2n=w n g n?||dw n||2g

At a maximum point P of w2n,w2n(P)=max V

n

w2n,?w2n≥0andθ(b u n)w2n=0.Hence: c u

n

(P)|w n(P)|≤g n(P)and

g n(P)≤ max R×V|?b?λ| |w n?1(P)|||du n(P)||g

Since c(λ,x)≥c0>0.We obtain the estimate for w n:

max

V|w n|≤1

?λ|)R(?)+(max R×V|

?c

c0 }max V|w n?1|

Then:

max

V|w n|≤

βR(?)+γ

R(ε)

(22) and we obtain for?small enough:

0<βR(?)+γ

c0?

f0

2

?g||?du||2g+||?2du||2g=g+g.(25) The relation(25)is a generalization of the classical relation(18).

3.2Regularity estimates

Proposition2Using the notations of Proposition(1),under assumptions1-2,there exists a ε>0such that the function of?,||?du?||∞,is bounded on any closed subinterval of[0, ε[. Proof:We take the exterior derivate of equation(23),multiply the result scalarly by ?duε.We get:and to the solution uεof equation(23)and use this equation:

εg+

?)u?],?duε>g+(?d(c u

ε

u?),?duε>g=g

To computeg,we apply the relation(25):ε

We have:

<θ(b u ε)?du ε,?du ε>g =g +g

<θ(b u ε)?du ε,?du ε>g =1

2

θ(b u ε)||?du ε||2g +

g

Finally we get the identity:

ε

2

θ(b u ε)||?du ε||2g +c u ε||?du ε||2g +S 1=S 2+S 3where:

S 1=g

S 2=g

S 3=g +g

To prove that ||?du ε||∞is bounded we follow the same method as the one used to prove the boundedness of ||du ε||∞.Let P ∈V be a point where the function:x ∈V –>||?du ε||2g (x )attains its maximum.Then ε2θ(b u ε)||?du ε||2g (P )=0.Hence:c u ε(P )||?du ε||2g (P )≤

3 n =1|S n (P )|(28)To estimate S 1,it is convenient to choose an orthonormal coframe ?eld ω1,...,ωm in a neighborhood of P.Denote by e 1,...,e m the corresponding frame ?eld.Then:

?du ε=U αβωα?ωβ

where the matrix of functions U αβis symmetric in α,β.Moreover,because U αβis sym-metric we can choose the coframe ?eld ω1,...,ωm so that at P:

U αβ(P )=0if α=β

?b =B αβe α?ωβ

B 1(?du ε)αβ=U αγB γβ+U βγB γαg =

m α,β,γ=1U αβ(U αγB γβ+U βγB γα)g (P )=m β,γ=1

U ββ(P )U γγ(P )(B γβ+B βγ

)(P )

|g (P )|≤b 0m

α=1U αα(P )2=b 0||?du ε||2g (P ).

To estimate the term B 2(b u ε,du ε),for ω∈T ?q V ,X,Y ∈T q V,q ∈V :

B 2(b u ε,du ε)[X,Y ]=du ε((?2b )u ε[X,Y ])+du ε

?X ?b ?λ u ε .du ε(X )+du ε ?b ?λ2

(u ε(q ),q ) .du ε(X ).du ε(Y ).Hence:

||B 2(b u ε,du ε)||∞

=||du ε||∞.||?2b ||∞+2||du ε||2∞. ??b ?λ2 ∞+β||du ε||∞.||?du ε||g (P )and |S 1(P )|≤b 0||?du ε||2g (P )+R (ε)||?du (P )||g {||?2b ||∞+2R (ε) ?

?b ?λ2 ∞+β||?du ε||g (P )}

||S 1||∞≤(b 0+βR (ε))||?du ε||2∞+BR (ε)[1+R (ε)]2||?du ε||∞(29)

where:

B =max {||?2b ||∞, ??b ?λ2 ∞

}Estimate of S 2(P):|S 2(P )|≤[K 1R (ε)||b ||∞+ε(K 2R (ε)+K 3||?du ε(P )||g )]||?du ε(P )||g ,

where the constants K 1and K 3depend only on the curvature of g and K 2on the covariant derivatives of the Ricci curvature tensor.

||S 2||∞≤[K 1R (ε)||b ||∞+ε(K 2R (ε)+K 3||?du ε||∞)]||?du ε||∞

(30)

Estimate of S 3(P):

|S 3(P )|≤||?du ||∞{||?d f ||∞+C (1+R (ε))2+|u (P )?c

The inequalities (28293031)imply the estimate:

c 0||?du ε||∞≤(b 0+βR (ε)+γ+εK 3)||?du ε||∞+[BR (ε)+C ](1+R (ε))2+

K 1R (ε)||b ||∞+εK 2R (ε)+||?d f ||∞(32)

The relation(22)implies that:

f 0

εor the

root of :εR (ε)=f 0ε]

—>R +is a decreasing function.So any sequence {u εn |εn ↓0as n ↑∞}contains a subsequence converging in the C 0-topology to a Lipschitz continuous function u such that sup V

||du ||g ≤R (0).Any limit u of u ε,as εtends to 0,satis?es the weak equation where the integrals are taken with respect to the volume measure associated to the metric g:?φ∈H 1(V ), V [u (θ(b u )φ)+c u uφ]= V fφ

高一英语人教版必修三unit1课文内容

Unit 1 Festivals around the world FESTIVALS AND CELEBRATIONS Festivals and celebrations of all kinds have been held everywhere since ancient times. Most ancient festivals would celebrate the end of cold weather, planting in spring and harvest in autumn. Sometimes celebrations would be held after hunters had caught animals. At that time people would starve if food was difficult to find, especially during the cold winter months. Today’s festivals have many origins, some religious, some seasonal, and some for special people or events. Festivals of the dead Some festivals are held to honour the dead or to satisfy the ancestors, who might return either to help or to do harm. For the Japanese festival Obon, people should go to clean graves and light incense in memory of their ancestors. They also light lamps and play music because they think that will lead the ancestors back to earth. In Mexico, people celebrate the Day of the Dead in early November. On this important feast day, people eat food in the shape of skulls and cakes with “bones” on them. They offer food, flowers and gifts to the dead. The Western holiday Halloween also had its origin in old beliefs about the return of the spirits of dead people. It is now a children’s festival, when they can dress up and go to their neighbours’ homes to ask for sweets. If the neighbours do not give any sweets, the children might play a trick on them. Festivals to Honour People Festivals can also be held to honour famous people. The Dragon Boat Festival in China honours the famous ancient poet, Qu Y uan. In the USA, Columbus Day is in memory of the arrival of Christopher Columbus in the New World. India has a national festival on October 2 to honour Mohandas Gandhi, the leader who helped gain India’s independence from Britain. Harvest Festivals Harvest and Thanksgiving festivals can be very happy events. People are grateful because their food is gathered for the winter and the agricultural work is over. In European countries, people will usually decorate churches and town halls with flowers and fruit, and will get together to have meals. Some people might win awards for their farm produce, like the biggest watermelon or the most handsome rooster. China and Japan have mid-autumn festivals when people admire the moon and in China enjoy moon-cakes. Spring Festivals The most energetic and important festivals are the ones that look forward to the end of winter and to the coming of spring. At the Spring Festival in China, people eat dumplings, fish and meat and may give children lucky money in red paper. There are dragon dances and carnivals, and families celebrate the Lunar New Year together. Some Western countries have very exciting carnivals, which take place forty days before Easter, usually in February. These carnivals might include parades, dancing in the streets day and night, loud music and colourful clothing of al kinds. Easter is an important religious and social festival for Christians around the world. It celebrates the return of Jesus from the dead and the coming of spring and new life. Japan’s Cherry Blossom Festival happens a little later. The country, covered with cherry tree flowers, looks as though it is covered with pink snow. People love to get together to eat, drink and have fun with each other. Festivals let us enjoy life, be proud of our customs and forget our work for a little while.

2019人教版高中英语必修3电子课本 word版

普通高中课程标准实验教科书《英语》电子课本 Book 3 Unit 1 Festivals around the world B3U1P1-3 FESTIV ALS AND CELEBRATIONS Ancient Festivals Festivals and celebrations of all kinds are held everywhere. The most ancient festivals would celebrate the end of the cold weather, planting in spring and harvest in autumn. Other celebrations were held when hunters could catch animals. They would starve if food was difficult to find, so they celebrated when they had food. They lit fires and made music because they thought these festivals would bring a year of plenty. Festivals of the Dead Some festivals are held to honour the dead, or satisfy and please the ancestors, who could return either to help or to do harm. In Japan the festival is called Obon, when people should go to clean the graves and light incense in memory of their ancestors. They light lamps and play music because they think that this will lead the ancestors back to earth. In Mexico they have the Day of the Dead in early November. On this important feast day, people might eat food in shape of skulls, and cakes with “bones” on them. They offer food, flowers and gifts to the dead. The festival of Halloween had its origin as an event in memory of the dead. It is now a children’s festival, when they can go to their neighbours’ homes and ask for sweets. They dress up and try to frighten people. If they are not given anything, the children might play a trick. Festivals to Honour People Festivals can be held as an honour to famous people or to the gods. One of these is the Dragon Boat Festival in China, which honours the famous ancient poet, Qu Yuan. Another is Columbus Day in the USA, in memory of the arrival of Christopher Columbus in America. In India there is a national festival on October 2 to honour Mahatma Gandhi, the leader who helped gain India’s independence from Britain. Harvest Festivals Harvest and Thanksgiving festivals can be very happy events. People are grateful because their food is gathered for the winter ,and because a season of agricultural work is over. In European countries it is the custom to decorate churches and town halls with flowers and fruit, and people get together to have meals. Some people might win awards for their animals, flowers, fruits and vegetables, like the biggest watermelon or the most handsome rooster. In China and Japan there are mid-autumn festivals, when people admire the moon and give gift of mooncakes. Spring Festivals The most energetic and important festivals are the ones that look forward to the end of winter and to the coming of spring. At the Spring Festival in China, people eat dumplings, fish and meat, and may give children lucky money in the red paper. There are dragon dances and carnivals, and families celebrate the lunar New Year together. In some Western countries there are very exciting carnivals, which take place forty days before Easter, usually in February. They might include parades, dancing in the streets day and night, loud music and colourful clothing of all kinds. Easter is an important religious and social festival in Christian countries. It celebrates the return of Jesus for Christians and it also celebrates the coming of spring. In Japan, the Cherry Blossom Festival happens a little later. The country is covered with cherry tree flowers so that it looks as though it might be covered with pink snow. People love to get together to eat, drink and have fun with each others. Festivals let us enjoy life, be proud of our customs and forget our daily life for a little while.

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(完整版)高中英语必修三教材分析_英语_教材分析_人教版

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烈的掌声感谢张主任真挚教诲! 我相信，我们在场的每一位老师听了讲座后，不仅仅是豁然开朗，更会紧跟秦主任的身后，努力去践行“老老实实做人、勤勤恳恳做事、认认真真做学问”，追寻终极目标，实现自己的人生价值! 最后，让我们再次以热烈的掌声，感谢张主任在百忙之中抽出时间为我们带来了这一席丰盛的精神大餐，同时我们也把掌声送给我们自己，为我们自己取得真经，得以提升而高兴!今天的活动到此结束!【篇二】讲座主持词开场白及结束语 亲爱的各位老师,各位同学,大家晚上好!欢迎大家参加****系列活动。 今天，我们有幸请到****(演讲人的主要头衔)为大家做一场关于“大学生，如何提高自身的核心竞争力”的讲座，相信这是很多同学感兴趣的话题。 首先请允许我代表广大师生对***到来表示热烈的欢迎和由衷的感谢! (接下来要写一小段对演讲人生平的简介，不要多，一两百字即可，挑最重头的，也是捧一捧嘉宾，能够查一下他近期有没有参加过于这个讲座内容相关的活动，也说两句，表示你做了功课，而且又给了嘉宾很好的引子，他如果有意可以在开场时顺着他最近参加的活动说，这样整个衔接都很流畅了) (然后是对讲座内容的简单引入，三、五句即可，用问

人教版高中英语必修3 unit3 完整课文原文

THE MILLION POUND BANK NOTE Act I，Scene3 NARRATOR: It is the summer of 1903.Two old and wealthy brothers, Roderick and Oliver, have made a bet.Oliver believes that with a million pound bank note a man could survive a month in London.His brother Roderick doubts it.At this moment, they see a penniless young man wandering on the pavement outside their house.It is Henry Adams, an American businessman, who is lost in London and does not know what he should do. RODERICK: Young man, would you step inside a moment, please? HENRY: Who? Me, sir? RODERICK: Yes, you. OLIVER: Through the front door on your left. HENRY: (A servant opens a door) Thanks. SERV ANT: Good morning, sir. Would you please come in? Permit me to lead the way,sir .OLIVER: (Henry enters) Thank you, James. That will be all. . RODERICK: How do you do, Mr ... er ...? HENRY: Adams. Henry Adams.

分层架构与业务逻辑实现方式

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理方面具有丰富的理论研究和实践经验。 围绕职业生涯规划，石勘教授将为我们进行首场讲座，让我们用热烈的掌声欢迎石勘教授为我们讲课。 感谢石勘教授!今天石勘教授针对当前青年就业创业，不同职业特点需求，职场要求和规划，结合案例为我们进行了深入、精彩的传授!相信会对我们如何找准职业规划定位，实现个人成才发展提供积极的指导和帮助。 我提议，让我们再次用热烈的掌声，感谢石勘教授的精彩授课! 同志们，今天是元宵节，团区委以及东升方圆集团的刘宝平先生专门在这里为大家准备了晚餐和元宵。 请大家在这里共进晚餐，共度元宵佳节。 晚餐时间是6：00，之前这段时间请大家参观“观光南瓜园”。 现在散会。

(完整word版)高中英语必修三课文详解Book3-unit3

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