Scaling Dynamics of a Massive Piston in a Cube Filled With Ideal Gas Exact Results
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Scaling Dynamics of a Massive Piston in a Cube Filled With Ideal Gas:Exact Results N.Chernov 1,4,J.L.Lebowitz 2,4,and Ya.Sinai 3February 1,2008Dedicated to Robert Dorfman on the occasion of his 65th birthday Abstract We continue the study of the time evolution of a system consisting of a piston in a cubical container of large size L ?lled with an ideal gas.The piston has mass M ～L 2and undergoes elastic collisions with N ～L 3gas particles of mass m .In a previous paper,Lebowitz,Piasecki and Sinai [LPS]considered a scaling regime,with time and space scaled by L ,in which they argued heuristically that the motion of the piston and the one particle distribution of the gas satisfy autonomous coupled di?erential equations.Here we state exact results and sketch proofs for this behavior.1The model and main results This paper is a continuation of [LPS],where deterministic scaled equations describing the dynamics of a massive piston in a cubical container ?lled with ideal gas were given.Here we state exact conditions on the validity of those equations and outline the arguments.Full proofs will be published in a separate paper [CLS].We refer the reader to [LPS]and to [CLS]as well as [G,GF,GP,KBM,Li]for a detailed description of the problem of a massive piston moving in a cylinder.Here we just recall necessary facts.Consider a cubical domain ΛL of size L separated into two parts by a wall (piston),which can move freely without friction inside ΛL .Each part of ΛL is ?lled by a noninter-
acting gas of particles,each of mass m .The piston has mass M =M L and moves along the x -axis under the action of elastic collisions with the particles.The size L of the cube
is a large parameter of the model,and we are interested in the limit behavior as L→∞. The mass m of gas particles is?xed.We will assume that M grows proportionally to L2 and the number of gas particles N is proportional to L3,while their velocities remain of order one.
The position of the piston at time t is speci?ed by a single coordinate X=X L(t), 0≤X≤L,its velocity is then given by V=V L(t)=˙X L(t).Since the components of the particle velocities perpendicular to the x-axis play no role in the dynamics,we may assume that each particle has only one coordinate,x,and one component of velocity,v, directed along the x-axis.
When a particle with velocity v hits the piston with velocity V,their velocities after the collision,v′and V′,respectively,are given by
V′=(1?ε)V+εv(1.1)
v′=?(1?ε)v+(2?ε)V(1.2) whereε=2m/(M+m).We assume that M+m=2mL2/a,where a>0is a constant,
so that
ε=2m
L2
(1.3)
When a particle collides with a wall at x=0or x=L,its velocity just changes sign.
The evolution of the system is completely deterministic,but one needs to specify initial conditions.We shall assume that the piston starts at the midpoint X L(0)=L/2 with zero velocity V L(0)=0.The initial con?guration of gas particles is chosen at random as a realization of a(two-dimensional)Poisson process on the(x,v)-plane(restricted to 0≤x≤L)with density L2p L(x,v),where p L(x,v)is a function satisfying certain conditions,see below,and the factor of L2is the cross-sectional area of the container.In other words,for any domain D?[0,L]×I R1the number of gas particles(x,v)∈D at time t=0has a Poisson distribution with parameterλD=L2 D p L(x,v)dx dv.
Let?L denote the space of all possible con?gurations of gas particles inΛL.For each realizationω∈?L the piston trajectory will be denoted by X L(t,ω)and its velocity by V L(t,ω).
As L→∞,space and time are rescaled as
y=x/L andτ=t/L.(1.4) which is a typical rescaling for the hydrodynamic limit transition(see[LPS,CLS]for motivation and physical discussion).We call y andτthe macroscopic(“slow”)variable, as opposed to the original microscopic(“fast”)x and t.Now let
Y L(τ,ω)=X L(τL,ω)/L,W L(τ,ω)=V L(τL,ω)(1.5) be the position and velocity of the piston in the macroscopic variables.The initial density p L(x,v)satis?es
p L(x,v)=π0(x/L,v)
where the functionπ0(y,v)is independent of L.Without loss of generality,assume that
π0is normalized so that 1
0 ∞
?∞
π0(y,v)dv dy=1
Then the mean number of particles in the entire containerΛL is equal exactly to
E(N)= L0 ∞?∞L2p L(x,v)dv dx=L3
In order to describe the dynamics by di?erential equations,we assume that the func-tionπ0(y,v)satis?es several technical requirements stated below.
(P1)Smoothness.π0(y,v)is a piecewise C1function with uniformly bounded partial derivatives,i.e.|?π0/?y|≤D1and|?π0/?v|≤D1for some D1>0.
(P2)Discontinuity lines.π0(y,v)may be discontinuous on the line y=Y L(0)(i.e.,“on the piston”).In addition,it may have a?nite number(≤K1)of other discontinuity lines in the(y,v)-plane with strictly positive slopes(each line is given by an equation v=f(y)where f(y)is C1and0 (P3)Density bounds.Let π0(y,v)>πmin>0for v1<|v| sup y,v π0(y,v)=πmax<∞(1.7) The requirements(1.6)and(1.7)basically mean thatπ0(y,v)takes values of order one. (P4)Velocity“cuto?”.Let π0(y,v)=0,if|v|≤v min or|v|≥v max(1.8) with some0 (P5)Approximate pressure balance.π0(y,v)must be nearly symmetric about the piston, i.e. |π0(y,v)?π0(1?y,?v)|<ε0(1.9) for all0 The requirements(P4)and(P5)are made to ensure that the piston velocity|V L(t,ω)| will be smaller than the minimum speed of the particles,with probability close to one, for times t=O(L).Such assumptions were?rst made in[LPS]. We think of D1,K1,c1,c2,v1,v2,v min,v max,πmin andπmax in(P1)–(P4)as?xed (global)constants andε0in(P5)as an adjustable small parameter.We will assume throughout the paper thatε0is small enough,meaning that ε0<ˉε0(D1,K1,c1,c2,v1,v2,v min,v max,πmin,πmax) It is important to note that the hydrodynamic limit does not require thatε0→0.The parameterε0stays positive and?xed as L→∞. Here is our main result: Theorem1.1There is an L-independent function Y(τ)de?ned for allτ≥0and a positiveτ?≈2/v max(actually,τ?→2/v max asε0→0),such that sup 0≤τ≤τ?|Y L(τ,ω)?Y(τ)|→0(1.10) and sup 0≤τ≤τ?|W L(τ,ω)?W(τ)|→0(1.11) in probability,as L→∞.Here W(τ)=˙Y(τ). This theorem establishes the convergence in probability of the random functions Y L(τ,ω),W(τ,ω)characterizing the mechanical evolution of the piston to the determin-istic functions Y(τ),W(τ),in the hydrodynamic limit L→∞. The functions Y(τ)and W(τ)satisfy certain(Euler-type)di?erential equations stated in the next section.Those equations have solutions for allτ≥0,but we can only guarantee the convergence(1.10)and(1.11)forτ<τ?.What happens forτ>τ?, especially asτ→∞,remains an open problem.Some experimental results and heuristic observations in this direction are presented in[CL]and discussed in Section4. Remark1.The function Y(τ)is at least C1and,furthermore,piecewise C2on the interval(0,τ?).Its?rst derivative W=˙Y(velocity)and its second derivative A=¨Y (acceleration)remainε0-small:supτ|W(τ)|≤const·ε0and supτ|A(τ)|≤const·ε0,see the next section. Remark2.We also estimate the speed of convergence in(1.10)and(1.11):there is aτ1>0(τ1≈1/v max)such that|Y L(τ,ω)?Y(τ)|=O(ln L/L)for0<τ<τ1and |Y L(τ,ω)?Y(τ)|=O(ln L/L1/7)forτ1<τ<τ?.The same bounds are valid for |W L(τ,ω)?W(τ)|.These estimates hold with“overwhelming”probability,speci?cally they hold for allω∈??L??L such that P(??L)=1?O(L?ln L). 2Hydrodynamical equations The equations describing the deterministic function Y(τ)involve another deterministic function–the density of the gasπ(y,v,τ).Initially,π(y,v,0)=π0(y,v),and forτ>0 the densityπ(y,v,τ)evolves according to the following rules. (H1)Free motion.Inside the container the density satis?es the standard continuity equation for a noninteracting particle system without external forces: ??y π(y,v,τ)=0(2.1) for all y except y=0,y=1and y=Y(τ). Equation(2.1)has a simple solution π(y,v,τ)=π(y?vs,v,τ?s)(2.2) for0 (H2)Collisions with the walls.At the walls y=0and y=1we have π(0,v,τ)=π(0,?v,τ)(2.3) π(1,v,τ)=π(1,?v,τ)(2.4) (H3)Collisions with the piston.At the piston y=Y(τ)we have π(Y(τ)?0,v,τ)=π(Y(τ)?0,2W(τ)?v,τ)for v π(Y(τ)+0,v,τ)=π(Y(τ)+0,2W(τ)?v,τ)for v>W(τ)(2.5) where v represents the velocity after the collision and2W(τ)?v that before the collision;here d W(τ)= (H4)Piston’s velocity.The velocity W=W(τ)of the piston must satisfy the equation ∞?∞(v?W)2sgn(v?W)q(v,τ;Y,W)dv=0(2.8) We also remark that forτ>0,when(2.5)holds, W(τ)= vπ(Y?0,v,τ)dv π(Y+0,v,τ)dv i.e.the piston’s velocity is the average of the nearby particle velocities on each side. The system of(hydrodynamical)equations(H1)–(H4)is now closed and,given ap-propriate initial conditions,should completely determine the functions Y(τ),W(τ)and p(y,v,τ)forτ>0.To specify the initial conditions,we set p(y,v,0)=π(y,v)and Y(0)=0.5.The initial velocity W(0)does not have to be speci?ed,it comes“for free”as the solution of the equation(2.8)at timeτ=0.It is easy to check that the initial speed|W(0)|will be smaller than v min,in fact W(0)→0asε0→0in(P5).Note that if the initial conditions atτ=0do not satisfy(2.3)–(2.5),there will be a discontinuity in p asτ→0(see also Remark4below). Equation(2.8)has a unique solution W as long as the piston interacts with some gas particles on both sides,i.e.as long as inf{v:π(Y+0,v,τ)>0}≤sup{v:π(Y?0,v,τ)>0} Indeed,the left hand side of(2.8)is a continuous and strictly monotonically decreasing function of W,and it takes both positive and negative values.The solution W(τ)may not be continuous inτ,though.But ifπ(y,v,τ)is piecewise C1and has a?nite number of discontinuity lines with positive slopes(as we require ofπ0(y,v)in Section1),then W(τ)will be continuous and piecewise di?erentiable. Remark3.One can easily check that the total mass M= π(y,v,τ)dv dy and the total kinetic energy2E= v2π(y,v,τ)dv dy remain constant along any solution of our system of equations(H1)–(H4).Also,the mass in the left and right part ofΛL separately remains constant.Equation(2.8)also preserves the total momentum of the gas vπ(y,v,τ)dv dy,but it changes due to collisions with the walls. Remark4.Previously,Lebowitz,Piasecki and Sinai[LPS]studied the piston dy-namics under essentially the same initial conditions as our(P1)–(P5).They argued heuristically that the piston dynamics could be approximated by certain deterministic equations in the original(microscopic)variables x and t.The deterministic equations found in[LPS]correspond to our(2.2)–(2.6)with obvious transformation back to the variables x,t,but our main equation(2.8)has a di?erent counterpart in the context of [LPS],which reads d Here X=X(t)and V=V(t)=˙X(t)denote the deterministic position and velocity of the piston andπ(x,v,t)the density of the gas(the constant a appeared in(1.3)).We refer to[LPS]for more details and a heuristic derivation of(2.9).Since(2.9),unlike our (2.8),is a di?erential equation,the initial velocity V(0)has to be speci?ed separately, and it is customary to set V(0)=0.Alternatively,one can set V(0)=W(0),see[CLS]. Equation(2.9)can be reduced to(2.8)in the limit L→∞as follows.One can show(we omit details)that(2.9)is a dissipative equation whose solution with any(small enough) initial condition V(0)converges to the solution of(2.8)during a t-time interval of length ～ln L.That interval has length～L?1ln L on theτaxis,and so it vanishes as L→∞, this is why we replace(2.9)with(2.8)and ignore the initial condition V(0)when working with the thermodynamic variablesτand y.The equation(2.9)is not used in this paper. We now describe the solution of our equations(H1)-(H4)in more detail.Assume that for someτ>0the gas densityπ(y,v,τ)satis?es the same requirements(P1)-(P4)as those imposed on the initial functionπ0(y,v)in Sect.1,with constants D′1,K′1,c′1,c′2, v′1,v′2,v′min,v′max,π′min andπ′max,whose values are not essential,but are independent of τ. We also assume an analogue of(P5),but this one is not so straightforward,since the piston does not have to stay at the middle point y=0.5at any timeτ>0.We require that |Y(τ)?0.5|<ε′0(2.10) and for any point(y,v)with v′min≤|v|≤v′max there is another point(y?,v?)“across the piston”(i.e.such that(y?Y(τ))(y??Y(τ))<0)satisfying |y+y??1|<ε′0,|v+v?|<ε′0(2.11) and |π(y,v,τ)?π(y?,v?,τ)|<ε′0(2.12) for some su?ciently smallε′0>0.Actually,the map(y,v)→(y?,v?),which we denote by Rτ,is one-to-one and will be explicitly constructed below.The constantε′0here,just likeε0in(P5),is assumed to be small enough,and moreover ε′0 We now derive elementary but important consequences of the above assumptions. Since the densityπ(y,v,τ)vanishes for|v| Q0W2?2Q1W+Q2=0(2.14) where Q0= sgn v·q(v,τ;Y)dv(2.15) Q1= v sgn v·q(v,τ;Y)dv(2.16) Q2= v2sgn v·q(v,τ;Y)dv(2.17) with Y=Y(τ).The integrals Q0,Q1,Q2have the following physical meaning: mQ0=m L?m R mQ1=p L?p R mQ2=2(e L?e R) where m L,p L,e L represent the total mass,momentum and energy of the incoming gas particles(per unit length)on the left hand side of the piston,and m R,p R,e R–those on the right hand side of it.The value Q2also represents the net pressure exerted on the piston by the gas as if the piston did not move.Of course,if Q2(τ)=0,then we must have W(τ)=0,which agrees with(2.14). Next,under the above requirements onπ(y,v,τ),the function q(v,τ;Y)is,in a certain sense,nearly symmetric in v about v=0(see[CLS]for details).This fact implies that Q0and Q2are small,more precisely max{|Q0|,|Q2|}≤C′ε0(2.18) where C′>0is a constant depending on the parameters D′1,K′1,etc.,but not onε0.At the same time,the assumption(P3)guarantees that Q1≥Q1,min>0(2.19) where Q1,min is a constant depending onπ′min,v′1,v′2,etc.,but not onε0. Ifε0is small enough,there is a unique root of the quadratic polynomial(2.14)on the interval(?v′min,v′min),which corresponds to the only solution of(2.8).Since this root is smaller,in absolute value,than the other root of(2.14),it can be expressed by Q1? Q0(2.20) W(τ)= where the sign before the radical is“?”,not“+”.Of course,(2.20)applies whenever Q0=0,while for Q0=0we simply have W(τ)=Q2/2Q1. Eqs.(2.18)-(2.20)imply an upper bound on the piston velocity:|W(τ)|≤B′ε0for some constant B′>0depending on D′1,K′1,etc.,but not onε0.A similar bound holds for the piston acceleration A(τ)=˙W(τ),since (dQ0/dτ)W2?2(dQ1/dτ)W+(dQ2/dτ) A(τ)= and|dQ i/dτ|=|(dQ i/dY)W|≤const·ε0,see[CLS]for more details. Next we consider the evolution of a point(y,v)in the domain G:={(y,v):0≤y≤1}under the rules(H1)–(H3),i.e.as it moves freely with constant velocity and collides elastically with the walls and the piston.Denote by(yτ,vτ)its position and velocity at timeτ≥0.Then(H1)translates into˙yτ=vτand˙vτ=0whenever yτ/∈{0,1,Y(τ)}, (H2)becomes(yτ+0,vτ+0)=(yτ?0,?vτ?0)whenever yτ?0∈{0,1},and(H3)gives (yτ+0,vτ+0)=(yτ?0,2W(τ)?vτ?0)(2.21) whenever yτ?0=Y(τ).Note that(2.21)corresponds to a special case of the mechanical collision rules(1.1)–(1.2)withε=0(equivalently,m=0).Hence the point(y,v)moves in G as if it was a gas particle with zero mass. The motion of points in(y,v)is described by a one-parameter family of transfor-mations Fτ:G→G de?ned by Fτ(y0,v0)=(yτ,vτ)forτ>0.We will also write F?τ(yτ,vτ)=(y0,v0).According to(H1)–(H3),the densityπ(y,v,τ)satis?es a simple equation π(yτ,vτ,τ)=π(F?τ(yτ,vτ),0)=π0(y0,v0)(2.22) for allτ≥0.Also,it is easy to see that for eachτ>0the map Fτis one-to-one and preserves area,i.e.det|DFτ(y,v)|=1. Now,because of(P4),the initial densityπ0(y,v)can only be positive in the region G+:={(y,v):0≤y≤1,v min≤|v|≤v max} hence we will restrict ourselves to points(y,v)∈G+only.At any timeτ>0,the images of those points will be con?ned to the region G+(τ):=Fτ(G+).In particular,π(y,v,τ)=0for(y,v)/∈G+(τ). The map Rτ:(y,v)→(y?,v?)involved in(2.11)and(2.12)can now be de?ned as Rτ=Fτ?R0?F?τ,where R0(y,v)=(1?y,?v)is a simple re?ection“across the piston”at timeτ=0. We now make an important observation.If a fast point(yτ,vτ)collides with a slow piston,|W(τ)|?|vτ|,they cannot recollide too soon:the point must travel to a wall, bounce o?it,and then travel back to the piston before it hits it again. Therefore,as long as(P1)–(P4)hold,the collisions of each moving point(yτ,vτ)∈G+(τ)with the piston occur at well separated time moments,which allows us to e?ec-tively count them.For(x,v)∈G+ N(y,v,τ)=#{s∈(0,τ):y s=Y(s),v s=W(s)} is the number of collisions of the point(y,v)with the piston during the interval(0,τ). For eachτ>0,we partition the region G+(τ)into subregions G+n(τ):={Fτ(y,v):(y,v)∈G+&N(y,v,τ)=n} so G+n(τ)is occupied by the points that at timeτhave experienced exactly n collisions with the piston during the interval(0,τ). Now,for each n≥1we de?neτn>0to be the?rst time when a point(yτ,vτ)∈G+(τ) experiences its(n+1)-st collision with the piston,i.e. τn=sup{τ>0:G+n+1(τ)=?} In particular,τ1>0is the earliest time when a point(yτ,vτ)∈G+(τ)experiences its ?rst recollision with the piston.Hence,no recollisions occur on the interval[0,τ1),and we call it the zero-recollision interval.Similarly,on the interval(τ1,τ2)no more than one recollision with the piston is possible for any point,and we call it the one-recollision interval. The time momentτ?mentioned in Theorem1.1is the earliest time when a point (yτ,vτ)∈G+(τ)either experiences its third collision with the piston or has its second collision with the piston given that the?rst one occurred afterτ1.Hence,τ?≤τ2,and actuallyτ?is very close toτ2,see below. The following theorem summarizes the properties of the solutions of the hydrody-namical equations(H1)–(H4). Theorem2.1([CLS])Let T>0be given.If the initial densityπ0(y,v)satis?es(P1)– (P5)with a su?ciently smallε0,then (a)the solution of our hydrodynamical equations(H1)–(H4)exists and is unique on the interval(0,T); (b)the densityπ(y,v,τ)satis?es conditions similar to(P1)–(P4)for all0<τ also satis?es(2.10)–(2.13); (c)The piston velocity and acceleration remain small,|W(τ)|=O(ε0)and|A(τ)|= O(ε0); (d)we have|τk?k/v max|=O(ε0)for all1≤k |τ??τ2|=O(ε0). Corollary2.2Ifε0=0,so that the initial densityπ0(y,v)is completely symmetric about the piston,the solution is trivial:Y(τ)≡0.5and W(τ)≡0for allτ>0. Lastly,we demonstrate the reason for our assumption that all the discontinuity curves of the initial densityπ0(y,v)must have positive slopes.It would be quite tempting to let π0(y,v)have more general discontinuity lines,e.g.allow it be smooth for v min<|v| Example.Suppose the initial densityπ0(y,v)has a horizontal discontinuity line v=v0 (say,v0=v min or v0=v max).After one interaction with the piston the image of this discontinuity line can oscillate up and down,due to the?uctuations of the piston accel-eration(Fig.1).As time goes on,this oscillating curve will“travel”to the wall and come back to the piston,experiencing some distortions on its way,caused by the di?erences in velocities of its points(Fig.1).When this curve comes back to the piston again,it may well have“turning points”where its tangent line is vertical,or even contain vertical segments of positive length.This produces unwanted singularities or even discontinuities of the piston velocity and acceleration.The same phenomena can also occur when a discontinuity line of the initial densityπ0(y,v)has a negative slope. 3Sketch of the argument Our proof of Theorem1.1is based on large deviation estimates for the Poisson random variable: Lemma3.1([CLS])Let X be a Poisson random variable with parameterλ>0.For any b>0there is a c>0such that for all0 √ λ)≤2e?cB2 This shows that the probabilities of large deviations rapidly decay,as they do for the Gaussian distribution. The principal step in our proof of Theorem1.1is the velocity decomposition scheme described next.Let V L(t,ω)be the velocity of the piston at time t≥0for a random con?guration of particlesω∈?L.Let?t>0be a small time increment.Then the law of elastic collision(1.1)implies V L(t+?t,ω)=(1?ε)k V L(t,ω)+ε k j=1(1?ε)k?j·v j(3.1) Here k=k(t,?t,ω)is the number of particles colliding with the piston during the time interval(t,t+?t),and v j are their velocities numbered in the order in which the particles collide. We rearrange the formula(3.1)as follows: V L(t+?t,ω)=(1?εk)V L(t,ω)+ε k j=1v j+χ(1)+χ(2)(3.2) where χ(1)=V L(t,ω)[(1?ε)k?1+εk] and χ(2)=ε k j=1v j[(1?ε)k?j?1] Let us assume that the?uctuations of the velocity V L(s,ω)on the interval(t,t+?t), are bounded by some quantityδV: |V L(s,ω)?V L(t,ω)|≤δV(3.3) sup s∈(t,t+?t) Consider two regions on the x,v plane: D1= (x,v):v?V L(t,ω)?(sgn v)δV?t,v min<|v| D2= (x,v):v?V L(t,ω)+(sgn v)δV?t,v min<|v| Each of them is a union of two trapezoids D i=D+i∪D?i,i=1,2,where D?i denotes the upper and D+i the lower trapezoid,see Fig.2. Note that D1?D2.The bound(3.3)implies that all the particles in the region D1 necessarily collide with the piston during the time interval(t,t+?t).Moreover,the trajectory of every point(x,v)∈D1hits the piston within time?t.The bound(3.3) also implies that all the particles actually colliding with the piston during the interval (t,t+?t)are contained in D2. Let us denote by k±r the number of particles in the regions D±r for r=1,2at time t. We also denote by k?the number of particles actually colliding with the piston“on the left”,and by k+that number“on the right”(of course,k?+k+=k).Due to the above observations,k±1≤k±≤k±2. Now,suppose that t+?t<τ1L.Then we show that for typical con?gurationsωthe particles in each domain D r,r=1,2,have never collided with the piston before.There-fore,their number,k±r,r=1,2,satis?es the laws of Poisson distribution,in particular, the large deviation estimate in Lemma3.1applies.This gives the bound(for typicalω) λ±1??k±1≤k±≤λ±2+?k±2(3.6) where λ±r=E(k±r)=L2 F?t L(D±r)p L(x,v)dx dv and F?t L corresponds to the action of F?τ=F?t/L in the original time-space coordinates x,t.The deviations?k±r in(3.6)can be adjusted by using Lemma3.1.The di?erence λ±2?λ±1is estimated by λ±2?λ±1=L2 F?t L(D±2\D±1)p L(x,v)dx dv≤const·L2δV?t By putting all these estimates together we get tight bounds on k in(3.2).Similarly we get bounds on k j=1v j in(3.2).The following is the?nal result of this analysis: V L(t+?t,ω)?V L(t,ω)=D(t,ω)?t+χ3(3.7) Here D(t,ω)=a[Q0V2L(t,ω)?2Q1V L(t,ω)+Q2](3.8) and Q0,Q1,Q2are de?ned similarly to(2.15)-(2.17),in which Y(τ)must be replaced by the actual piston position X L(t,ω)/L.The error termχ3in(3.7)is bounded by |χ3|≤const·ln L √ L (3.9) which corresponds to Brownian motion-type random?uctuations. The term D(t,ω)in(3.7)represents the main(“deterministic”)force acting on the piston.The termχ3describes random?uctuations of that force.When the piston velocity stabilizes,then the main force D should vanish,and an“equilibrium”velocity ˉV L (t,ω)will be established.The latter is the root of the equation D(t,ω)=0,which is ˉV L (t,ω)= Q1? Q0(3.10) The reason why V L(t,ω)converges toˉV L is that D is almost proportional toˉV L?V L,i.e. 0 |V L(t,ω)?ˉV L(t,ω)| Now,the piston coordinate Y L(τ,ω)=X L(τL,ω)/L is the solution of the di?erential equation ˙Y L =V L=ˉV L+χ4 where|χ4| The proof on the one-recollision interval(τ1,τ?)goes along the same lines.One major di?erence is that the number of particles k±r,r=1,2,in the domain D±r constructed in the velocity decomposition scheme is no longer a Poisson variable,so Lemma3.1does not apply directly. To handle this new situation,we pull the domain D±r back in time,as we did before. But now that pullback involves one interaction with the piston(corresponding to the?rst collision of the particles in D±r with the piston,which occurs during the zero-recollision interval0 Since the piston position and velocity at the moment of that?rst collision are random, the preimage of D±r will be a random domain.Its shape will depend on the piston velocity V(t,ω)during the zero-recollision interval0 A full proof of Theorem1.1is given in[CLS].At present,we do not know if this theorem can be extended beyond the critical timeτ?,this is an open question.Some other open problems are discussed in the next section. 4Discussion and open problems 1.The main goal of this work is to prove that under suitable initial conditions random ?uctuations in the motion of a massive piston are small and vanish in the thermodynamic limit.We are,however,able to control those?uctuations e?ectively only as long as the surrounding gas particles can be described by a Poisson process,i.e.during the zero-recollision interval0<τ<τ1.In that case the random?uctuations are bounded by const·L?1ln L,see Remark2after Theorem1.1.Up to the logarithmic factor,this bound is optimal,see[CL]and earlier estimates by Holley[H],D¨u rr et al.[DGL]. During the one-recollision intervalτ1<τ<τ2,the situation is di?erent.The proba-bility distribution of gas particles that have experienced one collision with the piston is no longer a Poisson process,it has intricate correlations.We are only able to show that random?uctuations remain bounded by L?1/7,see again Remark2.Perhaps,our bound is far from optimal,but our numerical experiments reported in[CL]show that random ?uctuations indeed grow during the one-recollision interval. We have tested numerically whether random?uctuations remained small after more than one recollision,i.e.at timesτ>τ2.We found that for some initialπ0they actually increased very rapidly,and we conjectured that the rate of increase was exponential in τ.We found,indeed,that at timesτ～log L the?uctuations became large even on a macroscopic scale,and then many unexpected phenomena occurred[CL]. Interestingly,the exponential growth of random?uctuations seems to be related to the instability of our hydrodynamical equations.We found that small perturbations of the initial densityπ0can grow exponentially inτunder certain conditions,matching the growth of random?uctuations of the piston motion in the mechanical model.We refer the reader to[CL]for further discussion and to our work in progress[CCLP]. 2.It is clear that in our model recollisions of gas particles with the piston have a very “destructive”e?ect on the dynamics in the system.However,we need to distinguish between two types of recollisions. We say that a recollision of a gas particle with the piston is long if the particle hits a wall x=0or x=L between the two consecutive collisions with the piston.Otherwise a recollision is said to be short.Long recollisions require some time,as the particle has to travel all the way to a wall,bounce o?it,and then travel back to the piston before it hits it again.Short recollisions can occur in rapid succession. We have imposed the velocity cut-o?(P4)in order to avoid any recollisions for at least some initial period of time(which we call the zero-recollision interval).More precisely, the upper bound v max guarantees the absence of long recollisions.Without it,we would have to deal with arbitrarily fast particles that dash between the piston and the wall very many times in any interval(0,τ).On the other hand,the lower bound v min was assumed to exclude short recollisions. There are good reasons to believe,though,that short recollisions may not be so de-structive for the piston dynamics.Indeed,let a particle experience two or more collisions with the piston in rapid succession(i.e.without hitting a wall in between).This can occur in two cases:(i)the particle’s velocity is very close to that of the piston,or(ii) the piston’s velocity changes very rapidly.The latter should be very unlikely,since the deterministic acceleration of the piston is very small,cf.Theorem2.1c.In case(i),the recollisions should have very little e?ect on the velocity of the piston according to the rule(1.1),so that they may be safely ignored,as it was done already in earlier studies [H,DGL]. We therefore expect that our results can be extended to velocity distributions without a cut-o?from zero,i.e.allowing v min=0. 3.In our paper,L plays a dual role:it parameterizes the mass of the piston(M～L2), and it represents the length of the container(0≤x≤L).This duality comes from our assumption that the container is a cube. However,our model is essentially one-dimensional,and the mass of the piston M and the length of the interval0≤x≤L can be treated as two independent parameters.In particular,we can assume that the container is in?nitely long in the x direction(so,that L is in?nite),but the mass of the piston is still?nite and given by M～L2.In this case there are no recollisions with the piston,as long as its velocity remains small.Hence,our zero-recollision interval is e?ectively in?nite.As a result,Theorem1.1can be extended to arbitrarily large times.Precisely,for any T>0we can prove the convergence in probability: P sup0≤τ≤T|Y L(τ,ω)?Y(τ)|≤C T ln L/L →1 and P sup0≤τ≤T|W L(τ,ω)?W(τ)|≤C T ln L/L →1 as L→∞,where C T>0is a constant and Y(τ)and W(τ)=˙Y(τ)are the solutions of the hydrodynamical equations described in Section2. 4.Along the same lines as above,we can assume that the container is d-dimensional with d≥3.Then the mass of the piston and the density of the particles are proportional to L d?1rather than L2. When d is large,the gas particles are very dense on the x,v plane.This leads to a much better control over?uctuations of the particle distribution and the piston trajectory. As a result,Theorem1.1can be extended to the k-recollision interval(τk,τk+1),where k≥1depends on d.It can be shown that for any k≥1there is a d k≥3such that for all d≥d k the convergence(1.10)and(1.11)holds withτ?=τk.Therefore,a higher dimensional piston is more stable than a lower dimensional one. It would be interesting to investigate other modi?cations of our model that lead to more stable regimes.For example,let the initial densityπ0(y,v)of the gas depend on the factor a=εL2in such a way thatπ0(y,v)=a?1ρ(y,v),whereρ(y,v)is a?xed function. Then the particle density grows as a→0.This is another way to increase the density of the particles,but without changing the dimension.One may expect a better control over random?uctuations in this case,too. Acknowledgements.N.Chernov was partially supported by NSF grant DMS-9732728. J.Lebowitz was partially supported by NSF grant DMR-9813268and by Air Force grant F49620-01-0154.Ya.Sinai was partially supported by NSF grant DMS-9706794.This work was completed when N.C.and J.L.stayed at the Institute for Advanced Study with partial support by NSF grant DMS-9729992. References [CCLP]E.Caglioti,N.Chernov,J.L.Lebowitz,E.Presutti,in preparation. [CLS]N.Chernov,J.L.Lebowitz,and Ya.Sinai,Dynamic of a massive piston in an ideal gas,preprint,available at https://www.wendangku.net/doc/149316995.html,/chernov/pubs/. [CL]N.Chernov,J.L.Lebowitz,Dynamics of a massive piston in an ideal gas: oscillatory motion and approach to equilibrium,this volume of the journal. [DGL] D.D¨u rr,S.Goldstein,and J.L.Lebowitz,A mechanical model of Brownian motion,Commun.Math.Phys.78(1981),507–530. [G]Ch.Gruber,Thermodynamics of systems with internal adiabatic constraints: time evolution of the adiabatic piston,Eur.J.Phys.20(1999),259–266. [GF]Ch.Gruber,L.Frachenberg,On the adiabatic properties of a stochastic adiabatic wall:Evolution,stationary non-equilibrium,and equilibrium states,Phys.A, 272(1999),392–428. [GP]Ch.Gruber,J.Piasecki,Stationary motion of the adiabatic piston,Phys.A268 (1999),412–423. [H]R.Holley,The motion of a heavy particle in an in?nite one dimensional gas of hard spheres,Z.Wahrschein.verw.Geb.17(1971),181–219. [KBM] E.Kestemont,C.Van den Broeck,and M.Mansour,The“adiabatic”piston: and yet it moves,Europhys.Lett.,49(2000),143–149. [KT] A.N.Kolmogorov,V.M.Tihomirov,ε-entropy andε-capacity of sets in function spaces,Amer.Math.Soc.Transl.17(1961),277–364. [LPS]J.L.Lebowitz,J.Piasecki,and Ya.Sinai,Scaling dynamics of a massive piston in an ideal gas.In:Hard ball systems and the Lorentz gas,217–227,Encycl. Math.Sci.,101,Springer,Berlin,2000. [Le]J.L.Lebowitz,Stationary nonequilibrium Gibbsian ensembles,Phys.Rev.114 (1959),1192–1202. [Li] E.Lieb,Some problems in statistical mechanics that I would like to see solved, Physica A263(1999),491–499.. Figure1:A horizontal discontinuity line(bottom)comes o?the piston as an oscillating curve(top). Figure2:Region D1is bounded by solid lines.Region D2is bounded by a dashed line. 从实践的角度探讨在日语教学中多媒体课件的应用 在今天中国的许多大学，为适应现代化，信息化的要求，建立了设备完善的适应多媒体教学的教室。许多学科的研究者及现场教员也积极致力于多媒体软件的开发和利用。在大学日语专业的教学工作中，教科书、磁带、粉笔为主流的传统教学方式差不多悄然向先进的教学手段而进展。 一、多媒体课件和精品课程的进展现状 然而，目前在专业日语教学中能够利用的教学软件并不多见。比如在中国大学日语的专业、第二外語用教科书常见的有《新编日语》（上海外语教育出版社）、《中日交流标准日本語》（初级、中级）（人民教育出版社）、《新编基础日语（初級、高級）》（上海译文出版社）、《大学日本语》（四川大学出版社）、《初级日语》《中级日语》（北京大学出版社）、《新世纪大学日语》（外语教学与研究出版社）、《综合日语》（北京大学出版社）、《新编日语教程》（华东理工大学出版社）《新编初级（中级）日本语》（吉林教育出版社）、《新大学日本语》（大连理工大学出版社）、《新大学日语》（高等教育出版社）、《现代日本语》（上海外语教育出版社）、《基础日语》（复旦大学出版社）等等。配套教材以录音磁带、教学参考、习题集为主。只有《中日交流標準日本語（初級上）》、《初級日语》、《新编日语教程》等少数教科书配备了多媒体DVD视听教材。 然而这些试听教材，有的内容为日语普及读物，并不适合专业外语课堂教学。比如《新版中日交流标准日本语（初级上）》，有的尽管DVD视听教材中有丰富的动画画面和语音练习。然而，课堂操作则花费时刻长，不利于教师重点指导，更加适合学生的课余练习。比如北京大学的《初级日语》等。在这种情形下，许多大学的日语专业致力于教材的自主开发。 其中，有些大学的还推出精品课程，取得了专门大成绩。比如天津外国语学院的《新编日语》多媒体精品课程为2007年被评为“国家级精品课”。目前已被南开大学外国语学院、成都理工大学日语系等全国40余所大学推广使用。 Unit1 Americans believe no one stands still. If you are not moving ahead, you are falling behind. This attitude results in a nation of people committed to researching, experimenting and exploring. Time is one of the two elements that Americans save carefully, the other being labor. "We are slaves to nothing but the clock,” it has been said. Time is treated as if it were something almost real. We budget it, save it, waste it, steal it, kill it, cut it, account for it; we also charge for it. It is a precious resource. Many people have a rather acute sense of the shortness of each lifetime. Once the sands have run out of a person’s hourglass, they cannot be replaced. We want every minute to count. A foreigner’s first impression of the U.S. is li kely to be that everyone is in a rush -- often under pressure. City people always appear to be hurrying to get where they are going, restlessly seeking attention in a store, or elbowing others as they try to complete their shopping. Racing through daytime meals is part of the pace 硕037班 刘文3110038020 2011/4/20交互式多模型仿真与分析IMM算法与GBP算法的比较，算法实现和运动目标跟踪仿真，IMM算法的特性分析 多源信息融合实验报告 交互式多模型仿真与分析 一、 算法综述 由于混合系统的结构是未知的或者随机突变的，在估计系统状态参数的同时还需要对系统的运动模式进行辨识，所以不可能通过建立起一个固定的模型对系统状态进行效果较好的估计。针对这一问题，多模型的估计方法提出通过一个模型集{}(),1,2,,j M m j r == 中不同模型的切换来匹配不同目标的运动或者同一目标不同阶段的运动，达到运动模式的实时辨识的效果。 目前主要的多模型方法包括一阶广义贝叶斯方法（BGP1），二阶广义贝叶斯方法（GPB2）以及交互式多模型方法等（IMM ）。这些多模型方法的共同点是基于马尔科夫链对各自的模型集进行切换或者融合，他们的主要设计流程如下图： M={m1,m2,...mk} K 时刻输入 值的形式 图一 多模型设计方法 其中，滤波器的重初始化方式是区分不同多模型算法的主要标准。由于多模型方法都是基于一个马尔科夫链来切换与模型的，对于元素为r 的模型集{}(),1,2,,j M m j r == ，从0时刻到k 时刻，其可能的模型切换轨迹为 120,12{,,}k i i i k trace k M m m m = ，其中k i k m 表示K-1到K 时刻，模型切换到第k i 个， k i 可取1,2,,r ，即0,k trace M 总共有k r 种可能。再令1 2 1 ,,,,k k i i i i μ+ 为K+1时刻经由轨迹0,k trace M 输入到第1k i +个模型滤波器的加权系数，则输入可以表示为 0,11 2 1 12|,,,,|,,,???k k trace k k k i M k k i i i i k k i i i x x μ++=?∑ 可见轨迹0,k trace M 的复杂度直接影响到算法计算量和存储量。虽然全轨迹的 新视野三版读写B2U4Text A College sweethearts 1I smile at my two lovely daughters and they seem so much more mature than we,their parents,when we were college sweethearts.Linda,who's21,had a boyfriend in her freshman year she thought she would marry,but they're not together anymore.Melissa,who's19,hasn't had a steady boyfriend yet.My daughters wonder when they will meet"The One",their great love.They think their father and I had a classic fairy-tale romance heading for marriage from the outset.Perhaps,they're right but it didn't seem so at the time.In a way, love just happens when you least expect it.Who would have thought that Butch and I would end up getting married to each other?He became my boyfriend because of my shallow agenda:I wanted a cute boyfriend! 2We met through my college roommate at the university cafeteria.That fateful night,I was merely curious,but for him I think it was love at first sight."You have beautiful eyes",he said as he gazed at my face.He kept staring at me all night long.I really wasn't that interested for two reasons.First,he looked like he was a really wild boy,maybe even dangerous.Second,although he was very cute,he seemed a little weird. 3Riding on his bicycle,he'd ride past my dorm as if"by accident"and pretend to be surprised to see me.I liked the attention but was cautious about his wild,dynamic personality.He had a charming way with words which would charm any girl.Fear came over me when I started to fall in love.His exciting"bad boy image"was just too tempting to resist.What was it that attracted me?I always had an excellent reputation.My concentration was solely on my studies to get superior grades.But for what?College is supposed to be a time of great learning and also some fun.I had nearly achieved a great education,and graduation was just one semester away.But I hadn't had any fun;my life was stale with no component of fun!I needed a boyfriend.Not just any boyfriend.He had to be cute.My goal that semester became: Be ambitious and grab the cutest boyfriend I can find. 4I worried what he'd think of me.True,we lived in a time when a dramatic shift in sexual attitudes was taking place,but I was a traditional girl who wasn't ready for the new ways that seemed common on campus.Butch looked superb!I was not immune to his personality,but I was scared.The night when he announced to the world that I was his girlfriend,I went along ?哈弗曼编码 A method for the construction of minimum-re-dundancy codes, 耿国华1数据结构1北京:高等教育出版社,2005:182—190 严蔚敏,吴伟民.数据结构(C语言版)[M].北京:清华大学出版社,1997. 冯桂,林其伟,陈东华.信息论与编码技术[M].北京:清华大学出版社,2007. 刘大有,唐海鹰,孙舒杨,等.数据结构[M].北京:高等教育出版社,2001 ?压缩实现 速度要求 为了让它(huffman.cpp)快速运行，同时不使用任何动态库，比如STL或者MFC。它压缩1M数据少于100ms（P3处理器，主频1G）。 压缩过程 压缩代码非常简单，首先用ASCII值初始化511个哈夫曼节点： CHuffmanNode nodes[511]; for(int nCount = 0; nCount < 256; nCount++) nodes[nCount].byAscii = nCount; 其次，计算在输入缓冲区数据中，每个ASCII码出现的频率： for(nCount = 0; nCount < nSrcLen; nCount++) nodes[pSrc[nCount]].nFrequency++; 然后，根据频率进行排序： qsort(nodes, 256, sizeof(CHuffmanNode), frequencyCompare); 哈夫曼树，获取每个ASCII码对应的位序列： int nNodeCount = GetHuffmanTree(nodes); 构造哈夫曼树 构造哈夫曼树非常简单，将所有的节点放到一个队列中，用一个节点替换两个频率最低的节点，新节点的频率就是这两个节点的频率之和。这样，新节点就是两个被替换节点的父 Unit 1 1学习外语是我一生中最艰苦也是最有意义的经历之一。虽然时常遭遇挫折，但却非常有价值。 2我学外语的经历始于初中的第一堂英语课。老师很慈祥耐心，时常表扬学生。由于这种积极的教学方法，我踊跃回答各种问题，从不怕答错。两年中，我的成绩一直名列前茅。 3到了高中后，我渴望继续学习英语。然而，高中时的经历与以前大不相同。以前，老师对所有的学生都很耐心，而新老师则总是惩罚答错的学生。每当有谁回答错了，她就会用长教鞭指着我们，上下挥舞大喊：“错！错！错！”没有多久，我便不再渴望回答问题了。我不仅失去了回答问题的乐趣，而且根本就不想再用英语说半个字。 4好在这种情况没持续多久。到了大学，我了解到所有学生必须上英语课。与高中老师不。大学英语老师非常耐心和蔼，而且从来不带教鞭！不过情况却远不尽如人意。由于班大，每堂课能轮到我回答的问题寥寥无几。上了几周课后，我还发现许多同学的英语说得比我要好得多。我开始产生一种畏惧感。虽然原因与高中时不同，但我却又一次不敢开口了。看来我的英语水平要永远停步不前了。 5直到几年后我有机会参加远程英语课程，情况才有所改善。这种课程的媒介是一台电脑、一条电话线和一个调制解调器。我很快配齐了必要的设备并跟一个朋友学会了电脑操作技术，于是我每周用5到7天在网上的虚拟课堂里学习英语。 6网上学习并不比普通的课堂学习容易。它需要花许多的时间，需要学习者专心自律，以跟上课程进度。我尽力达到课程的最低要求，并按时完成作业。 7我随时随地都在学习。不管去哪里，我都随身携带一本袖珍字典和笔记本，笔记本上记着我遇到的生词。我学习中出过许多错，有时是令人尴尬的错误。有时我会因挫折而哭泣，有时甚至想放弃。但我从未因别的同学英语说得比我快而感到畏惧，因为在电脑屏幕上作出回答之前，我可以根据自己的需要花时间去琢磨自己的想法。突然有一天我发现自己什么都懂了，更重要的是，我说起英语来灵活自如。尽管我还是常常出错，还有很多东西要学，但我已尝到了刻苦学习的甜头。 8学习外语对我来说是非常艰辛的经历，但它又无比珍贵。它不仅使我懂得了艰苦努力的意义，而且让我了解了不同的文化，让我以一种全新的思维去看待事物。学习一门外语最令人兴奋的收获是我能与更多的人交流。与人交谈是我最喜欢的一项活动，新的语言使我能与陌生人交往，参与他们的谈话，并建立新的难以忘怀的友谊。由于我已能说英语，别人讲英语时我不再茫然不解了。我能够参与其中，并结交朋友。我能与人交流，并能够弥合我所说的语言和所处的文化与他们的语言和文化之间的鸿沟。 III. 1. rewarding 2. communicate 3. access 4. embarrassing 5. positive 6. commitment 7. virtual 8. benefits 9. minimum 10. opportunities IV. 1. up 2. into 3. from 4. with 5. to 6. up 7. of 8. in 9. for 10.with V. 1.G 2.B 3.E 4.I 5.H 6.K 7.M 8.O 9.F 10.C Sentence Structure VI. 1. Universities in the east are better equipped, while those in the west are relatively poor. 2. Allan Clark kept talking the price up, while Wilkinson kept knocking it down. 3. The husband spent all his money drinking, while his wife saved all hers for the family. 4. Some guests spoke pleasantly and behaved politely, while others wee insulting and impolite. 5. Outwardly Sara was friendly towards all those concerned, while inwardly she was angry. VII. 1. Not only did Mr. Smith learn the Chinese language, but he also bridged the gap between his culture and ours. 2. Not only did we learn the technology through the online course, but we also learned to communicate with friends in English. 3. Not only did we lose all our money, but we also came close to losing our lives. LZ77压缩算法实验报告 一、实验内容 使用C++编程实现LZ77压缩算法的实现。 二、实验目的 用LZ77实现文件的压缩。 三、实验环境 1、软件环境：Visual C++ 6.0 2、编程语言：C++ 四、实验原理 LZ77 算法在某种意义上又可以称为“滑动窗口压缩”，这是由于该算法将一个虚拟的，可以跟随压缩进程滑动的窗口作为术语字典，要压缩的字符串如果在该窗口中出现，则输出其出现位置和长度。使用固定大小窗口进行术语匹配，而不是在所有已经编码的信息中匹配，是因为匹配算法的时间消耗往往很多，必须限制字典的大小才能保证算法的效率；随着压缩的进程滑动字典窗口，使其中总包含最近编码过的信息，是因为对大多数信息而言，要编码的字符串往往在最近的上下文中更容易找到匹配串。 五、LZ77算法的基本流程 1、从当前压缩位置开始，考察未编码的数据，并试图在滑动窗口中找出最长的匹 配字符串，如果找到，则进行步骤2，否则进行步骤3。 2、输出三元符号组( off, len, c )。其中off 为窗口中匹 配字符串相对窗口边 界的偏移，len 为可匹配的长度，c 为下一个字符。然后将窗口向后滑动len + 1 个字符，继续步骤1。 3、输出三元符号组( 0, 0, c )。其中c 为下一个字符。然后将窗口向后滑动 len + 1 个字符，继续步骤1。 六、源程序 /********************************************************************* * * Project description: * Lz77 compression/decompression algorithm. * *********************************************************************/ #include 新大学日语简明教程课文翻译 第21课 一、我的留学生活 我从去年12月开始学习日语。已经3个月了。每天大约学30个新单词。每天学15个左右的新汉字，但总记不住。假名已经基本记住了。 简单的会话还可以，但较难的还说不了。还不能用日语发表自己的意见。既不能很好地回答老师的提问，也看不懂日语的文章。短小、简单的信写得了，但长的信写不了。 来日本不久就迎来了新年。新年时，日本的少女们穿着美丽的和服，看上去就像新娘。非常冷的时候，还是有女孩子穿着裙子和袜子走在大街上。 我在日本的第一个新年过得很愉快，因此很开心。 现在学习忙，没什么时间玩，但周末常常运动，或骑车去公园玩。有时也邀朋友一起去。虽然我有国际驾照，但没钱，买不起车。没办法，需要的时候就向朋友借车。有几个朋友愿意借车给我。 二、一个房间变成三个 从前一直认为睡在褥子上的是日本人，美国人都睡床铺，可是听说近来纽约等大都市的年轻人不睡床铺，而是睡在褥子上，是不是突然讨厌起床铺了？ 日本人自古以来就睡在褥子上，那自有它的原因。人们都说日本人的房子小，从前，很少有人在自己的房间，一家人住在一个小房间里是常有的是，今天仍然有人过着这样的生活。 在仅有的一个房间哩，如果要摆下全家人的床铺，就不能在那里吃饭了。这一点，褥子很方便。早晨，不需要褥子的时候，可以收起来。在没有了褥子的房间放上桌子，当作饭厅吃早饭。来客人的话，就在那里喝茶；孩子放学回到家里，那房间就成了书房。而后，傍晚又成为饭厅。然后收起桌子，铺上褥子，又成为了全家人睡觉的地方。 如果是床铺的话，除了睡觉的房间，还需要吃饭的房间和书房等，但如果使用褥子，一个房间就可以有各种用途。 据说从前，在纽约等大都市的大学学习的学生也租得起很大的房间。但现在房租太贵，租不起了。只能住更便宜、更小的房间。因此，似乎开始使用睡觉时作床，白天折小能成为椅子的、方便的褥子。 新视野大学英语第一册Unit 1课文翻译 学习外语是我一生中最艰苦也是最有意义的经历之一。 虽然时常遭遇挫折，但却非常有价值。 我学外语的经历始于初中的第一堂英语课。 老师很慈祥耐心，时常表扬学生。 由于这种积极的教学方法，我踊跃回答各种问题，从不怕答错。 两年中，我的成绩一直名列前茅。 到了高中后，我渴望继续学习英语。然而，高中时的经历与以前大不相同。 以前，老师对所有的学生都很耐心，而新老师则总是惩罚答错的学生。 每当有谁回答错了，她就会用长教鞭指着我们，上下挥舞大喊：“错！错！错！” 没有多久，我便不再渴望回答问题了。 我不仅失去了回答问题的乐趣，而且根本就不想再用英语说半个字。 好在这种情况没持续多久。 到了大学，我了解到所有学生必须上英语课。 与高中老师不同，大学英语老师非常耐心和蔼，而且从来不带教鞭！ 不过情况却远不尽如人意。 由于班大，每堂课能轮到我回答的问题寥寥无几。 上了几周课后，我还发现许多同学的英语说得比我要好得多。 我开始产生一种畏惧感。 虽然原因与高中时不同，但我却又一次不敢开口了。 看来我的英语水平要永远停步不前了。 直到几年后我有机会参加远程英语课程，情况才有所改善。 这种课程的媒介是一台电脑、一条电话线和一个调制解调器。 我很快配齐了必要的设备并跟一个朋友学会了电脑操作技术，于是我每周用5到7天在网上的虚拟课堂里学习英语。 网上学习并不比普通的课堂学习容易。 它需要花许多的时间，需要学习者专心自律，以跟上课程进度。 我尽力达到课程的最低要求，并按时完成作业。 我随时随地都在学习。 不管去哪里，我都随身携带一本袖珍字典和笔记本，笔记本上记着我遇到的生词。 我学习中出过许多错，有时是令人尴尬的错误。 有时我会因挫折而哭泣，有时甚至想放弃。 但我从未因别的同学英语说得比我快而感到畏惧，因为在电脑屏幕上作出回答之前，我可以根据自己的需要花时间去琢磨自己的想法。 突然有一天我发现自己什么都懂了，更重要的是，我说起英语来灵活自如。 尽管我还是常常出错，还有很多东西要学，但我已尝到了刻苦学习的甜头。 学习外语对我来说是非常艰辛的经历，但它又无比珍贵。 它不仅使我懂得了艰苦努力的意义，而且让我了解了不同的文化，让我以一种全新的思维去看待事物。 学习一门外语最令人兴奋的收获是我能与更多的人交流。 与人交谈是我最喜欢的一项活动，新的语言使我能与陌生人交往，参与他们的谈话，并建立新的难以忘怀的友谊。 由于我已能说英语，别人讲英语时我不再茫然不解了。 我能够参与其中，并结交朋友。 实验名称：LZSS压缩算法实验报告 一、实验内容 使用Visual 6..0 C++编程实现LZ77压缩算法。 二、实验目的 用LZSS实现文件的压缩。 三、实验原理 LZSS压缩算法是词典编码无损压缩技术的一种。LZSS压缩算法的字典模型使用了自适应的方式，也就是说，将已经编码过的信息作为字典， 四、实验环境 1、软件环境：Visual C++ 6.0 2、编程语言：C++ 五、实验代码 #include -- need byte-swap function for big endian CPU */ #define READ_LE32(X) *(uint32_t *)(X) #define WRITE_LE32(X,Y) *(uint32_t *)(X) = (Y) /* this assumes sizeof(long)==4 */ typedef unsigned long uint32_t; /* text (input) size counter, code (output) size counter, and counter for reporting progress every 1K bytes */ static unsigned long g_text_size, g_code_size, g_print_count; /* ring buffer of size N, with extra F-1 bytes to facilitate string comparison */ static unsigned char g_ring_buffer[N + F - 1]; /* position and length of longest match; set by insert_node() */ static unsigned g_match_pos, g_match_len; /* left & right children & parent -- these constitute binary search tree */ static unsigned g_left_child[N + 1], g_right_child[N + 257], g_parent[N + 1]; /* input & output files */ static FILE *g_infile, *g_outfile; /***************************************************************************** initialize trees *****************************************************************************/ static void init_tree(void) { unsigned i; /* For i = 0 to N - 1, g_right_child[i] and g_left_child[i] will be the right and left children of node i. These nodes need not be initialized. Also, g_parent[i] is the parent of node i. These are initialized to NIL (= N), which stands for 'not used.' For i = 0 to 255, g_right_child[N + i + 1] is the root of the tree for strings that begin with character i. These are initialized to NIL. Note there are 256 trees. */ for(i = N + 1; i <= N + 256; i++) g_right_child[i] = NIL; for(i = 0; i < N; i++) g_parent[i] = NIL; } /***************************************************************************** Inserts string of length F, g_ring_buffer[r..r+F-1], into one of the trees (g_ring_buffer[r]'th tree) and returns the longest-match position and length via the global variables g_match_pos and g_match_len. If g_match_len = F, then removes the old node in favor of the new one, because the old one will be deleted sooner. 习惯与礼仪 我是个漫画家，对旁人细微的动作、不起眼的举止等抱有好奇。所以，我在国外只要做错一点什么，立刻会比旁人更为敏锐地感觉到那个国家的人们对此作出的反应。 譬如我多次看到过，欧美人和中国人见到我们日本人吸溜吸溜地出声喝汤而面露厌恶之色。过去，日本人坐在塌塌米上，在一张低矮的食案上用餐，餐具离嘴较远。所以，养成了把碗端至嘴边吸食的习惯。喝羹匙里的东西也象吸似的，声声作响。这并非哪一方文化高或低，只是各国的习惯、礼仪不同而已。 日本人坐在椅子上围桌用餐是1960年之后的事情。当时，还没有礼仪规矩，甚至有人盘着腿吃饭。外国人看见此景大概会一脸厌恶吧。 韩国女性就座时，单腿翘起。我认为这种姿势很美，但习惯于双膝跪坐的日本女性大概不以为然，而韩国女性恐怕也不认为跪坐为好。 日本等多数亚洲国家，常有人习惯在路上蹲着。欧美人会联想起狗排便的姿势而一脸厌恶。 日本人常常把手放在小孩的头上说“好可爱啊！”，而大部分外国人会不愿意。 如果向回教国家的人们劝食猪肉和酒，或用左手握手、递东西，会不受欢迎的。当然，饭菜也用右手抓着吃。只有从公用大盘往自己的小盘里分食用的公勺是用左手拿。一旦搞错，用黏糊糊的右手去拿， 会遭人厌恶。 在欧美，对不受欢迎的客人不说“请脱下外套”，所以电视剧中的侦探哥隆波总是穿着外套。访问日本家庭时，要在门厅外脱掉外套后进屋。穿到屋里会不受欢迎的。 这些习惯只要了解就不会出问题，如果因为不知道而遭厌恶、憎恨，实在心里难受。 过去，我曾用色彩图画和简短的文字画了一本《关键时刻的礼仪》（新潮文库）。如今越发希望用各国语言翻译这本书。以便能对在日本的外国人有所帮助。同时希望有朝一日以漫画的形式画一本“世界各国的习惯与礼仪”。 练习答案 ５、 （１）止める並んでいる見ているなる着色した （２）拾った入っていた行ったしまった始まっていた 新视野大学英语第三版第二册读写课文翻译 Unit 1 Text A 一堂难忘的英语课 1 如果我是唯一一个还在纠正小孩英语的家长，那么我儿子也许是对的。对他而言，我是一个乏味的怪物：一个他不得不听其教诲的父亲，一个还沉湎于语法规则的人，对此我儿子似乎颇为反感。 2 我觉得我是在最近偶遇我以前的一位学生时，才开始对这个问题认真起来的。这个学生刚从欧洲旅游回来。我满怀着诚挚期待问她：“欧洲之行如何？” 3 她点了三四下头，绞尽脑汁，苦苦寻找恰当的词语，然后惊呼：“真是，哇！” 4 没了。所有希腊文明和罗马建筑的辉煌居然囊括于一个浓缩的、不完整的语句之中！我的学生以“哇！”来表示她的惊叹，我只能以摇头表达比之更强烈的忧虑。 5 关于正确使用英语能力下降的问题，有许多不同的故事。学生的确本应该能够区分诸如their/there/they're之间的不同，或区别complimentary 跟complementary之间显而易见的差异。由于这些知识缺陷，他们承受着大部分不该承受的批评和指责，因为舆论认为他们应该学得更好。 6 学生并不笨，他们只是被周围所看到和听到的语言误导了。举例来说，杂货店的指示牌会把他们引向stationary（静止处），虽然便笺本、相册、和笔记本等真正的stationery（文具用品）并没有被钉在那儿。朋友和亲人常宣称They've just ate。实际上，他们应该说They've just eaten。因此，批评学生不合乎情理。 7 对这种缺乏语言功底而引起的负面指责应归咎于我们的学校。学校应对英语熟练程度制定出更高的标准。可相反，学校只教零星的语法，高级词汇更是少之又少。还有就是，学校的年轻教师显然缺乏这些重要的语言结构方面的知识，因为他们过去也没接触过。学校有责任教会年轻人进行有效的语言沟通，可他们并没把语言的基本框架——准确的语法和恰当的词汇——充分地传授给学生。 多媒体数据压缩实验报告 篇一：多媒体实验报告_文件压缩 课程设计报告 实验题目：文件压缩程序 姓名：指导教师：学院：计算机学院专业：计算机科学与技术学号： 提交报告时间：20年月日 四川大学 一，需求分析： 有两种形式的重复存在于计算机数据中，文件压缩程序就是对这两种重复进行了压 缩。 一种是短语形式的重复，即三个字节以上的重复，对于这种重复，压缩程序用两个数字：1.重复位置距当前压缩位置的距离；2.重复的长度，来表示这个重复，假设这两个数字各占一个字节，于是数据便得到了压缩。 第二种重复为单字节的重复，一个字节只有256种可能的取值，所以这种重复是必然的。给 256 种字节取值重新编码，使出现较多的字节使用较短的编码，出现较少的字节使用较长的编码，这样一来，变短的字节相对于变长的字节更多，文件的总长度就会减少，并且，字节使用比例越不均 匀，压缩比例就越大。 编码式压缩必须在短语式压缩之后进行，因为编码式压缩后，原先八位二进制值的字节就被破坏了，这样文件中短语式重复的倾向也会被破坏（除非先进行解码）。另外，短语式压缩后的结果：那些剩下的未被匹配的单、双字节和得到匹配的距离、长度值仍然具有取值分布不均匀性，因此，两种压缩方式的顺序不能变。 本程序设计只做了编码式压缩，采用Huffman编码进行压缩和解压缩。Huffman编码是一种可变长编码方式，是二叉树的一种特殊转化形式。编码的原理是：将使用次数多的代码转换成长度较短的代码，而使用次数少的可以使用较长的编码，并且保持编码的唯一可解性。根据 ascii 码文件中各 ascii 字符出现的频率情况创建 Huffman 树，再将各字符对应的哈夫曼编码写入文件中。同时，亦可根据对应的哈夫曼树，将哈夫曼编码文件解压成字符文件. 一、概要设计： 压缩过程的实现: 压缩过程的流程是清晰而简单的: 1. 创建 Huffman 树 2. 打开需压缩文件 3. 将需压缩文件中的每个 ascii 码对应的 huffman 编码按 bit 单位输出生成压缩文件压缩结束。 新视野大学英语1课文翻译 1下午好！作为校长，我非常自豪地欢迎你们来到这所大学。你们所取得的成就是你们自己多年努力的结果，也是你们的父母和老师们多年努力的结果。在这所大学里，我们承诺将使你们学有所成。 2在欢迎你们到来的这一刻，我想起自己高中毕业时的情景，还有妈妈为我和爸爸拍的合影。妈妈吩咐我们：“姿势自然点。”“等一等,”爸爸说，“把我递给他闹钟的情景拍下来。”在大学期间，那个闹钟每天早晨叫醒我。至今它还放在我办公室的桌子上。 3让我来告诉你们一些你们未必预料得到的事情。你们将会怀念以前的生活习惯，怀念父母曾经提醒你们要刻苦学习、取得佳绩。你们可能因为高中生活终于结束而喜极而泣，你们的父母也可能因为终于不用再给你们洗衣服而喜极而泣！但是要记住：未来是建立在过去扎实的基础上的。 4对你们而言，接下来的四年将会是无与伦比的一段时光。在这里，你们拥有丰富的资源：有来自全国各地的有趣的学生，有学识渊博又充满爱心的老师，有综合性图书馆，有完备的运动设施，还有针对不同兴趣的学生社团——从文科社团到理科社团、到社区服务等等。你们将自由地探索、学习新科目。你们要学着习惯点灯熬油，学着结交充满魅力的人，学着去追求新的爱好。我想鼓励你们充分利用这一特殊的经历，并用你们的干劲和热情去收获这一机会所带来的丰硕成果。 5有这么多课程可供选择，你可能会不知所措。你不可能选修所有的课程，但是要尽可能体验更多的课程！大学里有很多事情可做可学，每件事情都会为你提供不同视角来审视世界。如果我只能给你们一条选课建议的话，那就是：挑战自己！不要认为你早就了解自己对什么样的领域最感兴趣。选择一些你从未接触过的领域的课程。这样，你不仅会变得更加博学，而且更有可能发现一个你未曾想到的、能成就你未来的爱好。一个绝佳的例子就是时装设计师王薇薇，她最初学的是艺术史。随着时间的推移，王薇薇把艺术史研究和对时装的热爱结合起来，并将其转化为对设计的热情，从而使她成为全球闻名的设计师。 6在大学里，一下子拥有这么多新鲜体验可能不会总是令人愉快的。在你的宿舍楼里，住在你隔壁寝室的同学可能会反复播放同一首歌，令你头痛欲裂！你可能喜欢早起，而你的室友却是个夜猫子！尽管如此，你和你的室友仍然可能成 价值工程 置，是一项十分有意义的工作。另外恶意代码的检测和分析是一个长期的过程，应对其新的特征和发展趋势作进一步研究，建立完善的分析库。 参考文献： [1]CNCERT/CC.https://www.wendangku.net/doc/149316995.html,/publish/main/46/index.html. [2]LO R,LEVITTK,OL SSONN R.MFC:a malicious code filter [J].Computer and Security,1995,14(6):541-566. [3]KA SP ER SKY L.The evolution of technologies used to detect malicious code [M].Moscow:Kaspersky Lap,2007. [4]LC Briand,J Feng,Y Labiche.Experimenting with Genetic Algorithms and Coupling Measures to devise optimal integration test orders.Software Engineering with Computational Intelligence,Kluwer,2003. [5]Steven A.Hofmeyr,Stephanie Forrest,Anil Somayaji.Intrusion Detection using Sequences of System calls.Journal of Computer Security Vol,Jun.1998. [6]李华,刘智,覃征,张小松.基于行为分析和特征码的恶意代码检测技术[J].计算机应用研究，2011，28（3）：1127-1129. [7]刘威，刘鑫，杜振华.2010年我国恶意代码新特点的研究.第26次全国计算机安全学术交流会论文集，2011，（09）. [8]IDIKA N,MATHUR A P.A Survey of Malware Detection Techniques [R].Tehnical Report,Department of Computer Science,Purdue University,2007. 0引言 现有的压缩算法有很多种，但是都存在一定的局限性，比如：LZw [1]。主要是针对数据量较大的图像之类的进行压缩，不适合对简单报文的压缩。比如说，传输中有长度限制的数据，而实际传输的数据大于限制传输的数据长度，总体数据长度在100字节左右，此时使用一些流行算法反而达不到压缩的目的，甚至增大数据的长度。本文假设该批数据为纯数字数据，实现压缩并解压缩算法。 1数据压缩概念 数据压缩是指在不丢失信息的前提下，缩减数据量以减少存储空间，提高其传输、存储和处理效率的一种技术方法。或按照一定的算法对数据进行重新组织，减少数据的冗余和存储的空间。常用的压缩方式[2,3]有统计编码、预测编码、变换编码和混合编码等。统计编码包含哈夫曼编码、算术编码、游程编码、字典编码等。 2常见几种压缩算法的比较2.1霍夫曼编码压缩[4]：也是一种常用的压缩方法。其基本原理是频繁使用的数据用较短的代码代替，很少使用 的数据用较长的代码代替，每个数据的代码各不相同。这些代码都是二进制码，且码的长度是可变的。 2.2LZW 压缩方法[5,6]：LZW 压缩技术比其它大多数压缩技术都复杂，压缩效率也较高。其基本原理是把每一个第一次出现的字符串用一个数值来编码，在还原程序中再将这个数值还成原来的字符串，如用数值0x100代替字符串ccddeee"这样每当出现该字符串时，都用0x100代替，起到了压缩的作用。 3简单报文数据压缩算法及实现 3.1算法的基本思想数字0-9在内存中占用的位最 大为4bit ， 而一个字节有8个bit ，显然一个字节至少可以保存两个数字，而一个字符型的数字在内存中是占用一个字节的，那么就可以实现2:1的压缩，压缩算法有几种，比如，一个自己的高四位保存一个数字，低四位保存另外一个数字，或者，一组数字字符可以转换为一个n 字节的数值。N 为C 语言某种数值类型的所占的字节长度，本文讨论后一种算法的实现。 3.2算法步骤 ①确定一种C 语言的数值类型。 —————————————————————— —作者简介：安建梅（1981-），女，山西忻州人，助理实验室，研究方 向为软件开发与软交换技术；季松华（1978-），男，江苏 南通人，高级软件工程师，研究方向为软件开发。 数据快速压缩算法的研究以及C 语言实现 The Study of Data Compression and Encryption Algorithm and Realization with C Language 安建梅①AN Jian-mei ；季松华②JI Song-hua （①重庆文理学院软件工程学院，永川402160；②中信网络科技股份有限公司，重庆400000）（①The Software Engineering Institute of Chongqing University of Arts and Sciences ，Chongqing 402160，China ； ②CITIC Application Service Provider Co.，Ltd.，Chongqing 400000，China ） 摘要：压缩算法有很多种，但是对需要压缩到一定长度的简单的报文进行处理时，现有的算法不仅达不到目的，并且变得复杂， 本文针对目前一些企业的需要，实现了对简单报文的压缩加密，此算法不仅可以快速对几十上百位的数据进行压缩，而且通过不断 的优化，解决了由于各种情况引发的解密错误，在解密的过程中不会出现任何差错。 Abstract:Although,there are many kinds of compression algorithm,the need for encryption and compression of a length of a simple message processing,the existing algorithm is not only counterproductive,but also complicated.To some enterprises need,this paper realizes the simple message of compression and encryption.This algorithm can not only fast for tens of hundreds of data compression,but also,solve the various conditions triggered by decryption errors through continuous optimization;therefore,the decryption process does not appear in any error. 关键词：压缩；解压缩；数字字符；简单报文Key words:compression ；decompression ；encryption ；message 中图分类号：TP39文献标识码：A 文章编号：1006-4311（2012）35-0192-02 ·192·从实践的角度探讨在日语教学中多媒体课件的应用
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