Analytic expressions for the single particle energies with a quadrupole-quadrupole interact

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流力第二章作业

2.3 If a certain liquid weighs 8600N/m³, what are the values of its density, specific volume, and specific gravity relative to water at 15℃? Use Appendix A.2.22 A hydraulic lift of the type commonly used for greasing automobiles consists of a 280.00-mm-diameter ram that slides in a 280.18-mm-diameter cylinder (similar to Fig.P2.21), the annular space being filled with oil having a kinematic viscosity of 0.00042m²/s and specific gravity of 0.86 . If the rate of travel of the ram is 0.22m/s, find the frictional resistance when 2 m of the ram is engaged in the cylinder.2.30 (a) Derive an expression for capillary rise (or depression) between two vertical parallel plates. (b) How much would you expect 10℃water to raise (in mm) if the clean glass plates are separated by 1.2mm ?3.7.3 If a triangle of height d and base b is vertical and submerged in liquid with its vertex at the liquid surface, derive an expression for the depth to its center of pressure.3.7.9 In the following Fig. the rectangular flashboard MN shown in cross section (a=5.4m) is pivoted at B. (a) What must be the maximum height of B above Nif the flashboard is on the verge of tipping when the water surface rises to M? (b ) Ifthe flashboard is pivoted at the location determined in (a ) and the water surface is 1m below M , what are the reactions at B and N per m length of board perpendicular to the figure?3.9.9 A cylindrical bucket of 250 mm diameter and 400 mm high weighing 20.0Ncontains oil (s=0.80) to a depth of 180mm .(a)When placed to float in water ,whatwill be the immersion depth to the bottom of the bucket ? (b) What is the maximumvolume of oil the bucket can hold and still float?3.21 The cross section of a gate is shown in the following Fig. Its dimensionnormal to the plane of the paper is 8 m and its shape is such that 22.0y x . Thegate is pivoted about O . Develop analytic expressions in terms of the water depth yupstream of the gate for the following: (a ) horizontal force. (b ) vertical force. (c )clockwise moment acting on the gate. Compute (a ), (b ) and (c ) for the case wherethe water depth is 2.5 m.3.31 Refer to Sample Prob.3.10.suppose the velocity of the airplane is 220m/s, with all other data unchanged .What then would be the slope of the liquid surface in the tank ?4.3.1 Classify the following cases of flow as to whether they are steady or unsteady, uniform or nonuniform: (a) water flowing from a tilted pail; (b) flow from a rotating lawn sprinkler; (c) flow through the hose leading to the sprinkler; (d) natural stream during dry-weather flow; (e) a natural stream during flood; (f) flow in a city water-distribution main through a straight section of constant diameter with no side connections. (Note: There is room for legitimate argument in some of the above cases, which should stimulate independent thought.)4.13.1 A flow is defined by )4=w+tt=u. What is thev1(41(3),),1(2+=+velocity of flow at the point (3, 2, 4) at 2=t ? What is the acceleration at that point at 2=t ? Specify units in terms of L and T .Discussion:What are the conditions for the formula of hydrodynamic lubrication?5.2.4 From point 1, a 25-mm-diameter pipe runs horizontally under the floor and then a 12.5-mm-diameter line runs 1 m up the wall to point 2. To maintain a pressure of 300 kPa at point 2, when 15o C water is flowing at 0.5 L/s, what pressure must be provided at point 1? Neglect friction.Additional problem. A centrifugal water pump with a suction pipe is shown in following figure. Pump output is s /m 03.03=V Q , the diameter of suction pipe m m 150=d , the vacuum that pump can reach is O mH 8.6)/(2=g p v ρ, and all head losses in the suction pipe O mH h w 21=. Determine the utmost elevation e h from the pump shaft to the water surface on the pond.6.8 A reducing right-angled bend lies in a horizontal plane. Water enters from the west with a velocity of 3 m/s and a pressure of 30 kPa, and it leaves toward the north. The diameter at the entrance is 500 mm and at the exit it is 400 mm. Neglecting any friction loss, find the magnitude and direction of the resultant force on the bend.6.16 Assuming ideal flow in the horizontal plane in following figure, calculate the magnitude and direction of the resultant force on the stationary blade. Note that the jet (m/s 12=j V , mm 150=j D ) is divided by the splitter so that one-third of the water is diverted toward A.6.48 For a lawn sprinkler in the figure 6.14(see textbook), develop an expression for the runway speed ω in terms of h , r and 2β. This would occur if there were no mechanical friction or air resistance, i.e., zero torque.7.4.2 Oil(85.0=s and 2N.s/m 24.0=μ) in a 100-mm-diameter pipe flows with a velocity of 3.5m/s. What is the Reynolds number? Note that in the Reynolds number the significant length D L =.7.4.6 The dimensions of a model airplane are 1:30 those of its prototype. In planned test in a pressure wind tunnel the model will operate at the same speed and air temperature as the prototype. What pressure in the wind tunnel, relative to the atmospheric pressure, will make the Reynolds number the same?7.7.1 Use dimensional analysis to arrange the following groups into dimension parameters: (a) τ, V , ρ; (b) V ,L , ρ,σ. Use the MLT system.7.7.3 Use dimensional analysis to derive an expression for the powerdeveloped by engine in terms of the torque T and rotative speed ω.。

The Non-Thermal Sunyaev-Zel'dovich Effect in Clusters of Galaxies

a r X i v :a s t r o -p h /0211649v 1 29 N o v 2002Astronomy &Astrophysics manuscript no.sznt˙babFebruary 2,2008(DOI:will be inserted by hand later)The Non-Thermal Sunyaev–Zel’dovich Eﬀect in Clusters ofGalaxiesS.Colafrancesco 1,P.Marchegiani 2and E.Palladino 21INAF -Osservatorio Astronomico di Roma via Frascati 33,I-00040Monteporzio,Italy.Email:cola@coma.mporzio.astro.it 2Dipartimento di Fisica,Universit`a di Roma “La Sapienza”,Piazzale A.Moro 2,I-000Roma,Italy Received 21May 2002/Accepted 13August 2002Abstract.In this paper we provide an general derivation of the non-thermal Sunyaev-Zel’dovich (SZ)eﬀect in galaxy clusters which is exact in the Thomson limit to any approximation order in the optical depth τ.The general approach we use allows also to obtain an exact derivation of the thermal SZ eﬀect in a self-consistent framework.Such a general derivation is obtained using the full relativistic formalism and overcoming the limitations of the Kompaneets and of the single scattering approximations.We compare our exact results with those obtained at diﬀerent approximation orders in τand we give estimates of the precision ﬁt.We veriﬁed that the third order approximation yields a quite good description of the spectral distortion induced by the Comptonization of CMB photons in the cluster atmosphere.In our general derivation,we show that the spectral shape of the thermal and non-thermal SZ eﬀect depends not only from the frequency but also from the cluster parameters,like the electron pressure and the optical depth and from the energy spectrum of the electron population.We also show that the spatial distribution of the thermal and non-thermal SZ eﬀect in clusters depends on a combination of the cluster parameters and on the spectral features of the eﬀect.To have a consistent description of the SZ eﬀect in clusters containing non-thermal phenomena,we also evaluate in a consistent way -for the ﬁrst time -the total SZ eﬀect produced by a combination of thermal and non-thermal electron population residing in the same environment,like is the case in radio-halo clusters.In this context we show that the location of the zero of the total SZ eﬀect increases non-linearly with increasing values of the pressure ratio between the non-thermal and thermal electron populations,and its determination provides a unique way to determine the pressure of the relativistic particles residing in the cluster atmosphere.We discuss,in details,both the spectral and the spatial features of the total (thermal plus non-thermal)SZ eﬀect and we provide speciﬁc predictions for a well studied radio-halo cluster like A2163.Our general derivation allows also to discuss the overall SZ eﬀect produced by a combination of diﬀerent thermal populations residing in the cluster atmosphere.Such a general derivation of the SZ eﬀect allows to consider also the CMB Comptonization induced by several electron populations.In this context,wediscuss how the combined observations of the thermal and non-thermal SZ eﬀect and of the other non-thermalemission features occurring in clusters (radio-halo,hard X-ray and EUV excesses)provide relevant constraints onthe spectrum of the relativistic electron population and,in turn,on the presence and on the origin of non-thermalphenomena in galaxy clusters.We ﬁnally discuss how SZ experiments with high sensitivity and wide spectralcoverage,beyond the coming PLANCK satellite,can deﬁnitely probe the presence of a non-thermal SZ eﬀect ingalaxy clusters and disentangle this source of bias from the cosmologically relevant thermal SZ eﬀect.Key words.Cosmology:theory –Galaxies:clusters:general –Intergalactic medium –Radiation mechanism:thermal –Radiation mechanism:non-thermal –Cosmic microwave background1.IntroductionCompton scattering of the Cosmic Microwave Background (CMB)radiation by hot Intra Cluster (hereafter IC)elec-trons –the Sunyaev Zel’dovich eﬀect (Zel’dovich and Sunyaev 1969,Sunyaev &Zel’dovich 1972,1980)–is an im-portant process whose spectral imprint on the CMB can be used as a powerful astrophysical and cosmological probe (see Birkinshaw 1999for a review).Such a scattering produces a systematic shift of the CMB photons from the Rayleigh-Jeans (RJ)to the Wien side of the spectrum.2Sergio Colafrancesco et al.:Non-Thermal SZ eﬀect in clustersAn approximate description of the scattering of an isotropic Planckian radiationﬁeld by a non-relativistic Maxwellian electron population can be obtained by means of the solution of the Kompaneets(1957)equation(see, e.g.,Sunyaev&Zel’dovich1980).The resulting change in the spectral intensity,∆I th,due to the scattering of CMB photons by a thermal electron distribution can be written as(k B T0)3∆I th=2x e x+1(e x−1)2m e c2 dℓn e k B T e,(3) where n e and T e are the electron density and temperature of the IC gas,respectively,σT is the Thomson cross section,valid in the limit T e≫T0,k B is the Boltzmann constant and m e c2is the rest mass energy of the electron.The Comptonization parameter is proportional to the integral along the line of sightℓof the kinetic pressure,P th=n e k B T e, of the IC gas.Thus,the previous Eq.(3)can be written asσTy th=Sergio Colafrancesco et al.:Non-Thermal SZ eﬀect in clusters3 Nonetheless,in addition to the thermal IC gas,many galaxy clusters contain a population of relativistic electrons which produce a diﬀuse radio emission(radio halos and/or relics)via synchrotron radiation in a magnetized ICM(see,e.g.,Feretti2000for a recent observational review).The electrons which are responsible for the radio halo emission must have energies E e∼>a few GeV to radiate at frequenciesν∼>30MHz in order to reproduce the main properties of the observed radio halos(see,e.g.,Blasi&Colafrancesco1999;Colafrancesco&Mele2001,and references therein).A few nearby clusters also show the presence of an EUV/soft X-ray excess(Lieu et al.1999,Kaastra et al.1999, 2002;Bowyer2000)and of an hard X-ray excess(Fusco-Femiano et al.1999-2000;Rephaeli et al.1999;Kaastra et al.1999;Henriksen1999)over the thermal bremsstrahlung radiation.These emission excesses over the thermal X-ray emission may be produced either by Inverse Compton Scattering(hereafter ICS)of CMB photons oﬀan additional population of relativistic electrons or by a combination of thermal(reproducing the EUV excess,Lieu et al.2000) and suprathermal(reproducing the hard X-ray excess by non-thermal bremsstrahlung,Blasi et al.2000,Dogiel2000, Sarazin&Kempner2000)populations of distinct origins.However,since the ineﬃcient non-thermal bremsstrahlung mechanism would require a large energy input and thus imply an excessive heating of the IC gas-which is not observed -a more complex electron population is required toﬁt both the radio halo and the EUV/hard X-ray spectra of galaxy clusters(Petrosian2001).The high-energy part(E∼>1GeV)of such spectrum contains non-thermal electrons and produce synchrotron radio emission which canﬁt the observed radio-halo features:this is the best studied region of the non-thermal spectrum in galaxy clusters.Indeed,in many of the clusters in which the SZ eﬀect has been detected there is also evidence for radio halo sources(see Colafrancesco2002).So it is of interest to assess whether the detected SZ eﬀect is,in fact,mainly produced by the thermal electron population or there is a relevant contribution from other non-thermal electron populations.We will show speciﬁcally in this paper that each one of the electron populations which reside in the cluster atmosphere produces a distinct SZ eﬀect with peculiar spectral and spatial features.The description of the non-thermal SZ eﬀect produced by a single electron population with a non-thermal spectrum has been attempted by various authors(McKinnon et al.1991;Birkinshaw1999;Ensslin&Kaiser2000;Blasi et al. 2000).Several limits to the non-thermal SZ eﬀect are available in the literature(see,e.g.,Birkinshaw1999for a review) from observations of galaxy clusters which contain powerful radio halo sources(such as A2163)or radio galaxies(such as A426),but only a few detailed analysis of the results(in terms of putting limits to the non-thermal SZ eﬀect)have been possible so far.Only a single devoted search for the SZ eﬀect expected from a relativistic population of electrons in the lobes of bright radio galaxies has been attempted to date(McKinnon et al.1991).No signals were seen and a detailed spectralﬁt of the data to separate residual synchrotron and SZ eﬀect signals was not done,so that the limits on the SZ eﬀect do not strongly constrain the electron populations in the radio lobes.Also the problem of detecting the non-thermal SZ eﬀect in radio-halo clusters is likely to be severe because of the associated synchrotron radio emission. In fact,at low radio frequencies,such a synchrotron emission could easily dominate over the small negative signal produced by the SZ eﬀect.At higher frequencies there is in principle more chance to detect the non-thermal SZ eﬀect, but even here there are likely to be diﬃculties in separating the SZ eﬀect from theﬂat-spectrum component of the synchrotron emission(Birkinshaw1999).From the theoretical point of view,preliminary calculations(Birkinshaw1999,Ensslin&Kaiser2000,Blasi etal.2000)of the non-thermal SZ eﬀect have been carried out in the diﬀusion approximation(τ≪1),in the limit of single scattering and for a single non-thermal population of electrons.Speciﬁcally,Ensslin&Kaiser(2000)and Blasi et al.(2000)considered the SZ eﬀect produced,under the previous approximations,by a supra-thermal tail of the Maxwellian electron distribution claimed to exist in the Coma cluster and concluded that the SZ eﬀect,even though of small amplitude,could be measurable in the sub-mm region by the next coming PLANCK experiment. However,Petrosian(2001)showed that the suprathermal electron distribution faces with several crucial problems, the main being the large heating that such electrons would induce through Coulomb collisions in the ICM.The large energy input of the suprathermal distribution in the ICM of Coma would heat the IC gas up to unreasonably high temperatures,k B T e∼1016K,which are not observed.In addition,Colafrancesco(2002)noticed that dust obscuration does not allow any detection of the SZ signal from Coma at frequencies∼>600GHz.Matters are signiﬁcantly more complicated if the full relativistic formalism is used.However,this is necessary,sincemany galaxy clusters show extended radio halos and the electrons which produce the diﬀuse synchrotron radio emission are certainly highly relativistic so that the use of the Kompaneets approximation is invalid.Moreover,the presence of thermal and non-thermal electrons in the same location of the ICM renders the single scattering approximation and the single population approach unreasonable,so that the treatment of multiple scattering among diﬀerent electronic populations coexisting in the same cluster atmosphere is necessary to describe correctly the overall SZ eﬀect.Here we derive the spectral and spatial features of the SZ eﬀect using an exact derivation of the spectral distortion induced by a combination of a thermal and a non-thermal population of electrons which are present at the same time in the tely,we also consider the case of the combination of several non-thermal and thermal populations. The plan of the paper is the following:in Sect.2we will provide a general derivation of the SZ eﬀect in an exact formalism within the framework of our approach,i.e.,considering the fully relativistic approach outlined in Birkinshaw4Sergio Colafrancesco et al.:Non-Thermal SZ eﬀect in clusters(1999)and the eﬀect induced by multiple scattering.We work here in the Thomson limit hν≪m e c2.Within such framework,we derive the exact SZ eﬀect for a single electron population both in the thermal and non-thermal cases. In Sect.3we derive the SZ eﬀect produced by a single population of non-thermal electrons providing both exact and approximate(up to any order inτ)expressions for the non-thermal SZ eﬀect.In Sect.4we derive the total SZ eﬀect produced by a combination of two populations:a distribution of thermal electrons-like that responsible for the X-ray emission of galaxy clusters-and a non-thermal electron distribution-like the one responsible for the radio halo emission which is present in many galaxy clusters.Our general approach allows to derive the expression for the SZ eﬀect produced by the combination of any electronic population.So,in Sect.5we consider also the SZ eﬀect generated by the combination of two diﬀerent thermal populations.In Sect.6we discuss the spatial features associated to the presence of a non-thermal SZ eﬀect superposed to the thermal SZ eﬀect.In Sect.7we derive limits on the presence and amplitude of the non-thermal SZ eﬀect discussing speciﬁcally the cases of a few clusters in which there is evidence for the presence of non-thermal,high-energy electrons.We show how the possible detection of a non-thermal SZ eﬀect can set relevant constraints on the relative electron population.We summarize our results and discuss our conclusions in theﬁnal Sect.8.We use H0=50km s−1Mpc−1andΩ0=1throughout the paper unless otherwise speciﬁed. 2.The SZ eﬀect for galaxy clusters:a generalized approachIn this section we derive a generalized expression for the SZ eﬀect which is valid in the Thomson limit for a generic electron population in the relativistic limit and includes also the eﬀects of multiple scattering.First we consider an expansion in series for the distorted spectrum I(x)in terms of the optical depth,τ,of the electron population.Then, we consider an exact derivation of the spectral distortion using the Fourier Transform method already outlined in Birkinshaw(1999).An electron with momentum p=βγ,withβ=v/c andγ=E e/m e c2increases the frequencyνof a scatteredCMB photon on average by the factor t≡ν′/ν=43,whereν′andνare the photon frequencies after and beforethe scattering,respectively.Thus,in the Compton scattering against relativistic electrons(γ≫1)a CMB photon is eﬀectively removed from the CMB spectrum and is found at much higher frequencies.We work here in the Thomson limit,(in the electron’s rest frameγhν≪m e c2),which is valid for the interesting range of frequencies at which SZ observations are feasible.The redistribution function of the CMB photons scattered once by the IC electrons writes in the relativistic limit as, P1(s)= ∞0dpf e(p)P s(s;p),(5)where f e(p)is the electron momentum distribution and P s(s;p)is the redistribution function for a mono-energetic electron distribution,with s≡ln(t).An analytical expression for the redistribution function of CMB photons which suﬀer a single scattering,P s(s;p),has been given by Ensslin&Kaiser(2000).The expression for such a redistribution function has been also given analytically by Fargion et al.(1997)and by Sazonov&Sunyaev(2000).Once the function P1(s)is known,it is possible to evaluate the probability that a frequency change s is produced by a number n of repeated,multiple scattering.This is given by the repeated convolutionP n(s)= +∞−∞ds1...ds n−1P1(s1)...P1(s n−1)P1(s−s1−...−s n−1)≡P1(s)⊗...⊗P1(s)n times(6)(see Birkinshaw1999),where the symbol⊗indicates each convolution product.As a result,the location of the maximum of the function P n(s)moves towards higher values of s for higher values of n and the distribution P n(s) widens a-symmetrically towards high values of s giving thus higher probabilities to have large frequency shifts.The resulting total redistribution function P(s)can be written as the sum of all the functions P n(s),each one weighted by the probability that a CMB photon can suﬀer n scatterings,which is assumed to be Poissonian with expected valueτ:P(s)=+∞n=0e−ττn2τ2P2(s)+...=e−τ δ(s)+τP1(s)+1Sergio Colafrancesco et al.:Non-Thermal SZ eﬀect in clusters5 whereI0(x)=2(k B T0)3e x−1(9)is the incident CMB spectrum in terms of the a-dimensional frequency x.In the following we will derive the expression for the distorted spectrum usingﬁrst an expansion in series ofτand then the exact formulas obtained with the Fourier Transform(FT)method.2.1.High orderτ-expansionIn the calculation of the SZ eﬀect in galaxy clusters it is usual to use the expression of P(s)which is derived in the single scattering approximation and in the diﬀusion limit,τ≪1.In these limits,the distorted spectrum writes,in our formalism,as:I(x)=J0(x)+τ J1(x)−J0(x) ,(10) whereJ0(x)= +∞−∞I0(xe−s)P0(s)ds= +∞−∞I0(xe−s)δ(s)ds=I0(x),(11) J1(x)= +∞−∞I0(xe−s)P1(s)ds.(12)To evaluate the SZ distorted spectrum I(x)up to higher order inτ,we make use of the general expression of the series expansion of the function P(s)P(s)=+∞n=0a n(s)τn,(13)which can be written,using Eq.(7),asP(s)=+∞k=0(−τ)k k′!P k′(s).(14)The general n-th order term is obtained by selecting the terms in the double summation which contain the optical depthτup to the n-th power.These terms are obtained for k′=n−k,and provide the following expression for the series expansion coeﬃcients:a n(s)=nk=0(−1)k n!n k=0 n k (−1)k P n−k(s).(15)Inserting the coeﬃcients a n(s)given in Eq.(15)in Eq.(13)and this last one in Eq.(8),the resulting spectral distortion due to the SZ eﬀect can be written as:I(x)=+∞n=0b n(x)τn,(16)where the coeﬃcients b n(x)are given by b n(x)=16Sergio Colafrancesco et al.:Non-Thermal SZ eﬀect in clusters2.2.Exact derivation of the SZ eﬀectTheredistribution function P (s )can also be obtained in an exact form considering all the terms of the series expansion in Eq.(14).In fact,since the Fourier transform (hereafter FT)of a convolution product of two functions is equal to the product of the Fourier transforms of the two functions,the FT of P (s )writes as˜P(k )=e −τ 1+τ˜P 1(k )+12π +∞−∞˜P (k )e iks dk (21)which is obtained as the anti Fourier transform of ˜P(k )given in Eq.(19).To compare the exact calculations of the SZ spectral distortion with those obtained in the non-relativistic limit [see eqs.(1-2)as for the thermal case],it is useful to write the distorted spectrum in the form∆I (x )=2(k B T 0)3I 0 1e x −1=∆i (x )2(k B T 0)3.The Comptonization parameter y is deﬁned,in our general approach,in terms of the pressureP of the considered electron population:y =σT 1+p 2)with η=m e c 2/k B T e ,one writes the pressure asP th =n e k B T e(25)and it is easy from Eq.(24)to re-obtain the Compton parameter in Eq.(3)asy th =σTm e c 2(26)(we consider here,for simplicity,an isothermal cluster).The relativistically correct expression of the function ˜g (x )for a thermal population of electrons writes,at ﬁrst order in τ,as:˜g (x )=∆iτk B T ek B T e [j 1−j 0],(27)where j i ≡J i(hc )2k B T e (j 1−j 0)+16τ2(j 3−3j 2+3j 1−j 0) .(28)Notice that while the expression derived for ˜g (x )at ﬁrst order approximation in τis independent of τ,the expression for ˜g (x )at higher order approximation in τdepends directly on τ.This is even more the case for the exact expressionSergio Colafrancesco et al.:Non-Thermal SZ eﬀect in clusters 7of ˜g (x )given in the following.In fact,using the exact form of the function P (s )given in Eq.(22),it is possible to write the exact form of the function ˜g (x )as:˜g (x )=m e c 2τ +∞−∞i 0(xe −s )P (s )ds −i 0(x ) .(29)The expression of ˜g (x )approximated at ﬁrst order in τas given byEq.(27)is the one to compare directly with the expression of g (x )obtained from the Kompaneets (1957)equation,since both are evaluated under the assumption of single scattering suﬀered by a CMB photon against the IC electrons.Fig.1shows how the function ˜g (x )tends to g (x )for lower and lower IC gas temperatures T e .This conﬁrms that the distorted spectrum obtained from the Kompaneets equation is the non-relativistic limit of the exact spectrum.Notice that the function ˜g (x )given in Eq.(29)is the spectral shape of the SZ eﬀect obtained in the exact calculation while the function g (x )refers to the 1st order approximated case of a single,thermal,non-relativistic population of electrons.Fig.1.The function g (x )(solid line)is compared with the function ˜g (x )for thermal electron populations with k B T e =10(dot-dashed),5(dashes),3(long dashes)and 1(dotted)keV,respectively.It is worth to notice that while in the non-relativistic case it is possible to separate the spectral dependence of the eﬀect [which is contained in the function g (x )]from the dependence on the cluster parameters [which are contained in Compton parameter y th ,see eqs.(1-3)],this is no longer valid in the relativistic case in which the function J 1depends itself also on the cluster parameters.Speciﬁcally,for a thermal electron distribution,J 1depends non-linearly from the electron temperature T e through the function P 1(s ).This means that,even at ﬁrst order in τ,the spectral shape ˜g (x )of the SZ eﬀect depends on the cluster parameters,and mainly from the electron pressure P th .To evaluate the errors done by using the non-relativistic expression g (x )instead of the relativistic,correct function ˜g (x )given in Eq.(29)we calculate the fractional error ε=|[g (x )−˜g (x )]/˜g (x )|for thermal populations with k B T e =10,5,3,2and 1keV and for three representative frequencies (see Table 1).The fractional errors in Table 1tend to decrease systematically at each frequency with decreasing temperature.Note,however,that the error found in the high frequency region,x =15,is much higher than the errors found at lower frequencies and produces uncertainty levels of ∼>50%for T e ∼>5keV.This indicates that the high frequency region of the SZ eﬀect is more aﬀected by the relativistic corrections and by multiple scattering eﬀects.In Table 2we show the fractional error εdone when considering the exact calculation of the thermal SZ eﬀect and those at ﬁrst,second and third order approximations in τfor two values of the optical depth,τ=10−2and τ=10−3,as reported in the Table caption.Even for the highest cluster temperatures here considered,k B T e ∼20keV,the diﬀerence between the exact and approximated calculations is ∼<0.25%at the minimum of the SZ eﬀect (x ∼2.3)and is ∼<2.23%in the high-frequency tail (x ∼15),the two frequency ranges where the largest deviations are expected and could be measurable,in principle.For high-temperature clusters,the third-order approximated calculations of the SZ eﬀect ensures a precision ∼<2%at any interesting frequency.8Sergio Colafrancesco et al.:Non-Thermal SZ eﬀect in clustersk B T e=5keV k B T e=2keVx=2.33.681.5714.814.331.40x=1545.9923.87F irst order T hird orderτ=10−2x=2.30.220.120.27x=151.71k B T e=20keV0.240.23x=6.50.230.230.18τ=10−2x=2.30.020.70.02x=150.21k B T e=10keV0.010.01x=6.50.020.310.25Table2.The fractional errorεdone considering theﬁrst,second and third order approximation inτcompared with the exact calculation of the thermal SZ eﬀect is reported for three interesting values of the frequency x.The assumed values of k B T e and τare shown in theﬁrst column of the Table.The values ofεare given in units of10−2.Using the general,exact relativistic approach discussed in this section,we evaluate the frequency location of the zero of the thermal SZ eﬀect,x0,for diﬀerent cluster temperatures in the range2−20keV.The frequency location of x0depends on the IC gas temperature(or more generally on the IC gas pressure)and in Fig.2we compare the temperature dependence of x0evaluated atﬁrst order inτ(which actually does not depend onτ)with that evaluated in the exact approach for valuesτ=10−3andτ=10−2.The location of the null of the SZ eﬀect increases by ∼4.4%for k B T e up to∼20keV.Assuming a quadraticﬁt x0≈a+bθe+cθ2e,whereθe=k B T e/m e c2,we calculate the coeﬃcients a,b and c whichﬁt the temperature dependence of x0.These are given in Table3.In particular thebExactτ=0.0014.60593.8262−5.4895F irst order4.4304Sergio Colafrancesco et al.:Non-Thermal SZ eﬀect in clusters9Fig.2.The location of x0for a thermal electron population is shown as a function of k B T e.The exact calculation forτ=10−2 (solid line)and forτ=10−3(dashed line)are shown together with the approximated calculation toﬁrst order inτ(dotted line).while forτ=0.001one getsx0=3.830+4.453θe−3.128θ2e.(31) The remaining diﬀerences with our coeﬃcients given in Table3are due to the diﬀerent T e andτranges used for the quadraticﬁt.3.The SZ eﬀect generated from a non-thermal electron populationUsing the same general analytical approach derived in Sect.2above we derive here the exact spectral features of the SZ eﬀect produced by a single non-thermal population of electrons.Such an exact derivation has been not done so far. For such a population we consider here two diﬀerent phenomenological spectra:i)a single power-law energy spectrum, n rel=n0E−α,like that which is able toﬁt the radio-halo spectra of many clusters,and ii)a double power-law spectrum which is able toﬁt both the radio halo spectrum and the EUV and hard X-ray excess spectra observed in some nearby clusters(see Petrosian2001).Our formalism is,nonetheless,so general that it can be applied to any electron distribution so far considered toﬁt both the observed radio-halo spectra and those of the EUV/Hard X-ray excesses(see,e.g.,Sarazin1999,Blasi&Colafrancesco1999,Colafrancesco&Mele2001,Petrosian2001).The single power-law electron population is described by the momentum spectrumf e,rel(p;p1,p2,α)=A(p1,p2,α)p−α;p1≤p≤p2(32) where the normalization term A(p1,p2,α)is given by:(α−1)A(p1,p2,α)=10Sergio Colafrancesco et al.:Non-Thermal SZ eﬀect in clustersthe IC gas produced by Coulomb collisions of the relativistic electrons becomes larger than the bremsstrahlung cooling rate of the IC gas (see,e.g.,Petrosian 2001).This would produce unreasonably and unacceptably large heating of the IC gas.For this reason,we consider also a double power-law electron spectrum which has a ﬂatter slope below a critical value p cr :f e,rel (p ;p 1,p 2,p cr ,α1,α2)=K (p 1,p 2,p cr ,α1,α2) p −α1p 1<p <p cr p −α1+α2crp −α2p cr <p <p 2(34)with a normalization factor given by:K (p 1,p 2,p cr ,α1,α2)= p 1−α11−p 1−α1crα2−1 −1.(35)If p cr ∼400,the electron distribution of Eq.(34)withα1∼0.5at p ∼<p cr can be extended down to very low energies without violating any constraint set by the IC gas heating.Hence,we assume here the following parameter values:α1=0.5,α2=2.5,p cr =400e p 2→∞.We again consider p 1as a free parameter.A crucial quantity in the calculation of the non-thermal SZ eﬀect is given by the number density of relativistic electrons,n e,rel .The quantity n e,rel in galaxy clusters can be estimated from the radio halo spectrum intensity,but this estimate depends on the assumed value of the IC magnetic ﬁeld and on the model for the evolution of the radio-halo spectrum (see,e.g.Sarazin 1999for time-dependent models and Blasi &Colafrancesco 1999,Colafrancesco &Mele 2001for stationary models).For the sake of illustration,the radio halo ﬂux J νfor a power-law spectrum is,in fact,given by J ν∝n e,relB αr −1ν−αr ,where αr is the radio halo spectral slope.Because of the large intrinsic uncertainties existing in the density of the relativistic electrons which produce the cluster non-thermal phenomena,a value n e,rel =10−6cm −3for p 1=100has been assumed here.Our results on the amplitude of the non-thermal SZ eﬀect can be easily rescaled to diﬀerent values of n e,rel :in fact,decreasing (increasing)n e,rel will produce smaller (larger)amplitudes of the SZ eﬀect,as can be seen from eqs.(22,24,28,29).The density n e,rel increases for decreasing values of p 1.In fact,multiplying the electron distribution in Eq.(32)by the quantity n e,rel (p 1)one obtainsN e (p ;p 1)≡n e,rel (p 1)A (p 1)p −α(36)where the function A (p 1)is given by Eq.(33).Thus,the electron density scales asn e,rel (p 1)=n e,rel (˜p 1)A (˜p 1)3pv (p )m e c =n e m e c 2(α−1)1+p 2 α−22 p 1p 2(38)(see,e.g.,Ensslin &Kaiser 2000),where B x is the incomplete Beta functionB x (a,b )= x0t a −1(1−t )b −1dt(39)(see,e.g.,Abramowitz &Stegun 1965).The optical depth of the non-thermal electron population is given byτrel (p 1)=2·10−6A (˜p 1)10−6cm −3ℓK (p 1).(41)。

7. Strehl Ratio

The spherical wave coherence diameter determines the system resolution. The angular resolution looking up is λ/rosL, and that looking down is λ/ros0. For distant sources rosL ≈ r0, which allows ro to be used in resolution calculations looking towards space. For HV-21 turbulence r0 is about 5 cm at 0.5 μm. For a system looking down from 100 km above the earth the coherence diameter is 2.2 m at this wavelength, which is a factor of 44 times the resolution looking up. This is why the resolution of an imaging system looking down toward the earth is much greater than that of an imaging system on the ground looking at an object in space.
7.1 Strehl Ratio for Propagation Through Turbulence
The problem of determining the on-axis intensity of a plane wave propagating through atmospheric turbulence was analyzed by Fried (1966), and the resultant integral was evaluated numerically. Here a compact analytic expression for the Strehl ratio is obtained. A second form expressible as an inﬁnite series is also obtained. This second solution is more convenient, since a few series terms give an accurate answer for small levels of turbulence. An asymptotic series is obtained that is applicable when turbulence is more severe. It is shown that the ﬁrst two terms of the series give an accurate result in the regime in which most astronomical telescopes operate.

光伏电站的电网改造和继电保护规划与设计

Grid-connected photovoltaic power systems: Technical and potential problems—A review?Renewable and Sustainable Energy ReviewsTraditional electric power systems are designed in large part to utilize large baseload power plants, with limited ability to rapidly ramp output or reduce output below a certain level. The increase in demand variability created by intermittent sources such as photovoltaic (PV) presents new challenges to increase system flexibility. This paper aims to investigate and emphasize the importance of the grid-connected PV system regarding the intermittent nature of renewable generation, and the characterization of PV generation with regard to grid code compliance. The investigation was conducted to critically review the literature on expected potential problems associated with high penetration levels and islanding prevention methods of grid tied PV. According to the survey, PV grid connection inverters have fairly good performance. They have high conversion efficiency and power factor exceeding 90% for wide operating range, while maintaining current harmonics THD less than 5%. Numerous large-scale projects are currently being commissioned, with more planned for the near future. Prices of both PV and balance of system components (BOS) are decreasing which will lead to further increase in use. The technical requirements from the utility power system side need to be satisfied to ensure the safety of the PV installer and the reliability of the utility grid. Identifying the technical requirements for grid interconnection and solving the interconnect problems such as islanding detection, harmonic distortion requirements and electromagnetic interference are therefore very important issues for widespread application of PV systems. The control circuit also provides sufficient control and protection functions like maximum power tracking, inverter current control and power factor control. Reliability, life span and maintenance needs should be certified through the long-term operation of PV system. Further reduction of cost, size and weight is required for more utilization of PV systems. Using PV inverters with a variable power factor at high penetration levels may increase the number of balanced conditions and subsequently increase the probability of islanding. It is strongly recommended that PV inverters should be operated at unity power factor.A 24-h forecast of solar irradiance using artificial neural network: Application for performance prediction of a grid-connected PV plant at Trieste, Italy?Solar EnergyForecasting of solar irradiance is in general significant for planning the operations of power plants which convert renewable energies into electricity. In particular, the possibility to predict the solar irradiance (up to 24?h or even more) can became – with reference to the Grid Connected Photovoltaic Plants (GCPV) – fundamental in making power dispatching plans and – with reference to stand alone and hybridsystems – also a useful reference for improving the control algorithms of charge controllers. In this paper, a practical method for solar irradiance forecast using artificial neural network (ANN) is presented. The proposed Multilayer Perceptron MLP-model makes it possible to forecast the solar irradiance on a base of 24?h using the present values of the mean daily solar irradiance and air temperature. An experimental database of solar irradiance and air temperature data (from July 1st 2008 to May 23rd 2009 and from November 23rd 2009 to January 24th 2010) has been used. The database has been collected in Trieste (latitude 45°40′N, longitude 13°46′E), Italy. In order to check the generalization capability of the MLP-forecaster, a K-fold cross-validation was carried out. The results indicate that the proposed model performs well, while the correlation coefficient is in the range 98–99% for sunny days and 94–96% for cloudy days. As an application, the comparison between the forecasted one and the energy produced by the GCPV plant installed on the rooftop of the municipality of Trieste shows the goodness of the proposed model.A new simple analytical method for calculating the optimum inverter size in grid-connected PV plants??Electric Power Systems ResearchA new simple analytical method for the calculation of the optimum inverter size in grid-connected PV plants in any location is presented. The derived analytical expressions contain only four unknown parameters, three of which are related to the inverter and one is related to the location and to the nominal power of the PV plant. All four parameters can be easily estimated from data provided by the inverter manufacturer and from freely available climate data. Additionally, analytical expressions for the calculation of the annual energy injected into the ac grid for a given PV plant with given inverter, are also provided. Moreover, an expression for the effective annual efficiency of an inverter is given. The analytical method presented here can be a valuable tool to design engineers for comparing different inverters without having to perform multiple simulations, as is the present situation. The validity of the proposed analytical model was tested through comparison with results obtained by detailed simulations and with measured data.Analysis of isolated power systems for village electrification?Energy for Sustainable DevelopmentA large part of the world's population, particularly in India and Africa, lives in villages that often lie beyond the reach of grid power supply. Isolated power systems, which generate power at site, are considered as a viable option for the electrification of these areas. This paper discusses Indian experiences of isolated power systems. In India, there are many villages which have been electrified through renewable isolated power plants like biomass gasifier and solar photovoltaic (PV)systems. Case studies have been conducted for three such isolated power plants in the state of Maharashtra, India. It is observed from these case studies that the existing power plants are oversized and have a potential for reduction in distribution losses. This paper proposes an integrated design method for isolated power system, which combines load modeling, sizing and optimum distribution network. The levelized unit cost of energy can be reduced by 25–50% for the case studies by adopting the integrated design methodology. Generic guidelines are evolved for systems design from the case studies of sample isolated power systems.Sustainable electricity generation for rural and peri-urban populations of sub-Saharan Africa: The “flexy-energy” concept??Energy PolicyDesign and load management Optimization are big concerns for hybrid systems. ? Hybrid solar PV/Diesel is economically viable for remote areas and environmental friendly. ? “Flexy-energy” concept is a flexible hybrid solar PV/diesel/biomass suitable for remote areas. ? “Flexy-energy” concept is a flexible hybrid solar PV/diesel/biomass suitable for remote areas.Multi-objective optimization of batteries and hydrogen storage technologies for remote photovoltaic systems??EnergyStand-alone photovoltaic (PV) systems comprise one of the promising electrification solutions to cover the demand of remote consumers, especially when it is coupled with a storage solution that would both increase the productivity of power plants and reduce the areas dedicated to energy production.This paper presents a multi-objective design of weakly connected systems simultaneously minimizing the total levelized cost and the connection to the grid, while fulfilling a constraint of consumer satisfaction.For this task, a multi-objective code based on particle swarm optimization has been used to find the best combination of different energy devices. Both short and mid terms based on forecasts assumptions have been investigated.An application for the site of La Nouvelle in the French overseas island of La R éunion is proposed. It points up a strong cost advantage by using lead-acid (Pb-A) batteries in the short term and a mitigated solution for the mid term between Pb-A batteries and Gaseous hydrogen (GH2). These choices depend on the cost, the occupied area and the local pollution and, of course, legislation.太阳方向跟踪系统The effects on grid matching and ramping requirements, of single and distributed PV systems employing various fixed and sun-tracking technologies??In this second paper, which studies the hourly generation data from the Israel Electric Corporation for the year 2006, with a view to adding very large-scale photovoltaic power (VLS-PV) plants, three major extensions are made to the results reported in our first paper. In the first extension, PV system simulations are extended to include the cases of 1- and 2-axis sun-tracking, and 2-axis concentrator photovoltaic (CPV) technologies. Secondly, the effect of distributing VLS-PV plants among 8 Negev locations, for which hourly metrological data exist, is studied. Thirdly, in addition to studying the effect of VLS-PV on grid penetration, the present paper studies its effect on grid ramping requirements. The principal results are as follows: (i) sun-tracking improves grid matching at high but not low levels of grid flexibility; (ii) geographical distribution has little effect on grid penetration; (iii) VLS-PV significantly increases grid ramping requirements, particularly for CPV systems, but not beyond existing ramping capabilities; (iv) geographical distribution considerably ameliorates this effect.Performance prediction of 20?kWp grid-connected photovoltaic plant at Trieste (Italy) using artificial neural network?基于人工智能网络的20千瓦太阳能入网接入系统设计Energy Conversion and Management 节能与电网智能管理学报Growing of PV for electricity generation is one of the highest in the field of the renewable energies and this tendency is expected to continue in the next years. Due to the various seasonal, hourly and daily changes in climate, it is relatively difficult to find a suitable analytic model for predicting the performance of a grid-connected photovoltaic (GCPV) plant. In this paper, an artificial neural network is used for modelling and predicting the power produced by a 20?kWp GCPV plant installed on the roof top of the municipality of Trieste (latitude 45°40′N, longitude 13°46′E), Italy. An experimental database of climate (irradiance and air temperature) and electrical (power delivered to the grid) data from January 29th to May 25th 2009 has been used. Two ANN models have been developed and implemented on experimental climate and electrical data. The first one is a multivariate model based on the solar irradiance and the air temperature, while the second one is an univariate model which uses as input parameter only the solar irradiance. A database of 3437 patterns has been divided into two sets: the first (2989 patterns) is used for training the different ANN models, while the second (459 patterns) is used for testing and validating the proposed ANN models. Prediction performance measures such as correlation coefficient (r) and mean bias error (MBE) are presented. The results show that good effectiveness is obtained between the measured and predicted power produced by the 20?kWp GCPV plant. In fact, the found correlation coefficient is in the range 98–99%, while the mean bias error varies between 3.1% and 5.4%.A techno-economic comparison of rural electrification based on solar home systems and PV microgrids太阳能电站入网的技术-经济效益分析对比农村电气化项目家庭用太阳能发电系统社区用太阳能微电网设计Solar home systems are typically used for providing basic electricity services to rural households that are not connected to electric grid. Off-grid PV power plants with their own distribution network (micro/minigrids) are also being considered for rural electrification. A techno-economic comparison of the two options to facilitate a choice between them is presented in this study on the basis of annualised life cycle costs (ALCC) for same type of loads and load patterns for varying number of households and varying length and costs of distribution network. The results highlight that microgrid is generally a more economic option for a village having a flat geographic terrain and more than 500 densely located households using 3–4 low power appliances (e.g. 9?W CFLs) for an average of 4?h daily. The study analyses the viability of the two options from the perspectives of the user, an energy service company and the society.。

Systematic risk and international portfolio choice

THE JOURNAL OF FINANCE•VOL.LIX,NO.6•DECEMBER2004Systemic Risk and International Portfolio Choice SANJIV RANJAN DAS and RAMAN UPPAL∗ABSTRACTReturns on international equities are characterized by jumps;moreover,these jumpstend to occur at the same time across countries leading to systemic risk.We capturethese stylized facts using a multivariate system of jump-diffusion processes where thearrival of jumps is simultaneous across assets.We then determine an investor’s opti-mal portfolio for this model of returns.Systemic risk has two effects:One,it reducesthe gains from diversification and two,it penalizes investors for holding levered posi-tions.We find that the loss resulting from diminished diversification is small,whilethat from holding very highly levered positions is large.R ETURNS ON INTERNATIONAL EQUITIES are characterized by jumps;1moreover,these jumps tend to occur at the same time across countries,implying that conditional correlations between international equity returns tend to be higher in periods of high market volatility or following large downside moves.2Our objective in ∗Sanjiv Ranjan Das and Raman Uppal are with Santa Clara University,and CEPR and London Business School,respectively.We are grateful to Stephen Lynagh for excellent research assistance. We thank an anonymous referee,Andrew Ang,Pierluigi Balduzzi,Suleyman Basak,Greg Bauer, Geert Bekaert,Harjoat Bhamra,Michael Brandt,John Campbell,George Chacko,Jo˜ao Cocco, Glen Donaldson,Darrell Duffie,Bernard Dumas,Ken Froot,Francisco Gomes,Eric Jacquier,Tim Johnson,Andrew Karolyi,Andrew Lo,Michelle Lee,Jan Mahrt-Smith,Scott Mayfield,Narayan Naik,Vasant Naik,Roberto Rigobon,Geert Rouwenhorst,Piet Sercu,Milind Shrikhande,Rob Stambaugh,Raghu Sundaram,Luis Viceira,Jiang Wang,Tan Wang,Greg Willard,Wei Xiong,and Hongjun Yan for their suggestions.We also gratefully acknowledge comments from seminar par-ticipants at Bank of England,Boston College,Erasmus University,HEC Paris,London Business School,Maryland University,MIT,Swiss National Bank,University of Essex,University of Exeter, University of Maryland,University of Rochester,the1999Global Derivatives Conference,VIIth International Conference on Stochastic Programming,International Finance Conference at Geor-gia Institute of Technology,Conference of International Association of Financial Engineers,NBER Finance Lunch Seminar Series,the1999meetings of the European Finance Association,the2001 meetings of the Western Finance Association,the2002meetings of INQUIRE,the2003meetings of the American Finance Association,and the2003CIRANO Conference on Portfolio Choice.1Evidence on jumps in international equity returns is provided by Jorion(1988),Akgiray and Booth(1988),Bates(1996),and Bekaert et al.(1998).2For example,on July19,2002,the Dow fell by4.6%,the Dax by5.0%,the Cac by5.4%,the FTSE by4.6%,and the Nikkei by2.8%.Similarly,world equity markets fell in lockstep on October27, 1997,when the drop from the12-month peak was9.2%in Britain,35.4%in Hong Kong,21.3% in Japan,2.1%in Australia,10.7%in Mexico,27.9%in Brazil,and9.1%in the United States. Other events with large correlated price drops include the Debt crisis of1982,the Mexican crisis in December1994,and the Russian crisis in August1998;see Rigobon(2003)for a complete list of dates with large market moves.For evidence on changing conditional correlations see,for instance, Speidell and Sappenfield(1992),Odier and Solnik(1993),Erb,Harvey,and Viskanta(1994),Longin and Solnik(1995),Karolyi and Stulz(1996),Chakrabarti and Roll(2000),and Ang and Chen(2002).28092810The Journal of Financethis paper is to evaluate the effect on portfolio choice of systemic risk,defined as the risk from infrequent events that are highly correlated across a large number of assets.Our contribution is to provide a mathematical model of security returns that captures the stylized facts about international equity returns described above. We do this by modeling security returns as jump-diffusion processes where jumps across assets are systemic(occur simultaneously),though the size of each jump is allowed to differ across assets.Next,we derive the optimal portfo-lio weights for this model of returns.Then,we calibrate the portfolio model to the U.S.equity index and to five international equity indexes.For robustness, we consider two sets of international indexes:the first for developed countries and the second for emerging countries.Systemic risk has two effects:It reduces the gains from diversification and also penalizes investors for holding levered positions.We find that the loss resulting from diminished diversification is small,while that from holding highly levered positions is large.For instance, the certainty equivalent cost of ignoring systemic risk for a conservative agent with relative risk aversion of3who is investing$1,000for1year is on the or-der of$0.10for the developed-country indexes and$3for the emerging-country indexes.However,for more aggressive investors who hold heavily levered port-folios,the cost of ignoring systemic risk is substantial:For example,an investor with a risk aversion of1who ignores systemic risk has a positive probability of losing all her wealth.Our work can be distinguished from the literature on portfolio choice with idiosyncratic jumps in returns,for example,Aase(1984),Jeanblanc-Picque and Pontier(1990),and Shirakawa(1990).In more recent work,Liu,Longstaff,and Pan(2003)study a model of portfolio choice with event risk.In contrast to these theoretical models,our motivation is to evaluate the effect of systemic jumps on portfolio selection by empirically estimating the parameters of the returns in our model,and implementing the model based on these estimates. In contrast to the static model in Chunhachinda et al.(1997),where polynomial goal programming is used to examine the effect of skewness on portfolio choice by assuming a utility function defined over the moments of the distribution of returns,our model is dynamic,with preferences given by a standard constant relative risk-averse utility function,and in our model the effect of skewness (and higher moments)arises because of jumps in the returns process rather than being introduced explicitly through the utility function.3Our work is also related to Ang and Bekaert(2002),who embed an inter-national portfolio choice problem in a dynamic model with a regime-switching data-generating process.Two regimes are considered that correspond to a 3For early work on how skewness influences portfolio choice,see Samuelson(1970),Tsiang (1972),and Kane(1982).Kraus and Litzenberger(1976)show the implications for equilibrium prices of a preference for positive skewness,while Kraus and Litzenberger(1983)derive the suf-ficient conditions on return distributions to get a three-moment(mean,variance,and skewness) capital asset pricing model.Harvey and Siddique(2000)provide an empirical test of the effect of skewness on asset prices.Systemic Risk and International Portfolio Choice2811 normal regime with low correlations and a downturn regime with higher cor-relations.In their setup,regimes can be persistent,and their paper includes an analysis of portfolio choice when the short interest rate and earnings yields predict returns.The framework in Ang and Bekaert,however,does not accom-modate intermediate consumption,admits only a numerical solution even in the absence of intermediate consumption,and is difficult to estimate when there are more than two regimes or three risky assets.In contrast to Ang and Bekaert,we develop a theoretical framework along the lines of Merton(1971);because our model nests the well-understood Merton model as a special case,it allows one to interpret cleanly the effect of systemic jumps.Also,we provide analytic expressions for the optimal portfolio weights. Moreover,our model can incorporate intermediate consumption(the solution to the portfolio problem stays the same),and can be estimated and implemented for any number of assets.While in the paper we consider only a simple IID en-vironment where the unconditional correlations between returns are constant over time,we argue that this is sufficient to show that the effect of systemic jumps will not be large even in the presence of regime shifts.4Our framework can also allow for predictability in returns and for other state variables,just as they are incorporated in Merton,but because this is tangential to our main objective,we do not include these features in the model we present.The rest of the paper is organized as follows.In Section I,we develop a model of asset returns that captures systemic risk.In Section II,we derive the opti-mal portfolio weights when asset returns have a systemic-jump component.In Section III,we describe our data,give the moments for the returns in the pres-ence of systemic risk,and estimate the parameters of the returns processes. We then calibrate the portfolio model to the estimated parameters in order to compare the portfolio weights of an investor who accounts for systemic risk and an investor who ignores systemic risk.We conclude in Section IV.Proofs for propositions are presented in the Appendix.I.Asset Returns with Systemic RiskIn this section,we develop a model of asset prices that allows for systemic jumps and compare it to a pure-diffusion model without jumps.The two features of the data that we wish our returns-model to capture are(1)large changes in asset prices,and(2)a high degree of correlation across these changes.To allow for large changes in returns,we introduce a jump component in prices;to model these jumps as being systemic,we assume that this jump is common across all assets,though the distribution of the jump size is allowed to vary across assets. We start by describing the standard continuous-time process that is typically assumed for asset returns:dS n=ˆαn dt+ˆσn dz n,n=1,...,N,(1)S n4Das and Uppal(2003)show how to extend the model to allow for persistence in jumps.2812The Journal of FinancewithE tdS nS n=ˆαn dt(2)E tdS nS n×dS mS m=ˆσnm dt=ˆσnˆσmˆρnm dt,(3)where S n is the price of asset n,N is the total number of risky assets being considered for the portfolio,and the correlation between the shocks dz n and dz m is denoted byˆρnm dt=E(dz n×dz m).We will denote the N×N matrix of the covariance terms arising from the diffusion components by Σ,with its typ-ical element beingˆσnm≡ˆσnˆσmˆρnm.We adopt the convention of denoting vectors and matrices with boldface characters in order to distinguish them from scalar quantities;parameters of the pure-diffusion returns process,and other quan-tities related to the pure-diffusion model,are denoted with aˆ(carat)over the variable.To allow for the possibility of infrequent but large changes in asset returns,5 we extend the specification in equation(1)by introducing a jump component to the process for returns,as in Merton(1976).In order to capture the systemic nature of these jumps,we impose two restrictions on the jump-diffusion pro-cesses:one,the jump is assumed to arrive at the same time across all risky assets;two,conditional on a jump,the jump size is assumed to be perfectly correlated across assets;that is,the value of all the assets jumps in the same direction.Below,we formally describe a returns process for the risky assets that has these properties.Introducing a jump component to the process of returns in(1),we have dS nS n=αn dt+σn dz n+(˜J n−1)dQ(λ),n=1,...,N,(4)where Q is a Poisson process with intensityλ,and(˜J n−1)is the random jump amplitude that determines the percentage change in the asset price if the Poisson event occurs.Given our desire to model the large changes in prices as occurring at the same time across the risky assets,we have assumed that the ar-rival of jumps is coincident across all risky assets;that is,dQ n(λn)=dQ m(λm)= dQ(λ),∀n={1,...,N},m={1,...,N}.6We also assume that the diffusion shock,the Poisson jump,and the random variable˜J n are independent and that J n≡ln(˜J n)has a normal distribution with meanµn and varianceν2n,implying that the distribution of the jump size is asset-specific(below we will assume5In contrast to systemic risk,systematic risk refers to correlation between assets and a common factor,but does not require that the size of this correlation be large or that the correlated changes be infrequent.6The returns process described above is IID;in particular,and in contrast to Ang and Bekaert (2002),there is no persistence in jumps.Das and Uppal(2003)show how one could extend this model to allow for persistence in systemic jumps by making the arrival rate of jumps,λ,stochastic.Systemic Risk and International Portfolio Choice 2813that,conditional on a jump,the jump sizes for different assets are perfectly correlated).Thus,for the process in (4),the total expected return in equation (5)has two components:One part comes from the diffusion process,αn and the other,denoted αJ n ,comes from the jump process:E t dS n S n=αn dt +αJ n dt .(5)We also assume that the jump size is perfectly correlated across assets;as we shall see in Section III,this turns out to be a conservative assumption,and has the further advantage of reducing the number of parameters to be estimated.The total covariance between dS n and dS m ,given in the equation below ,E t dS n S n × dS m S m=σnm dt +σJ nm dt ,(6)arises from two sources:The covariance between the diffusion components of the returns,σnm ≡σn σm ρnm ,and the covariance between the jump compo-nents,σJ nm .The N ×N matrix containing the covariation arising from the jump terms is denoted by J ,while the N ×N matrix of the covariance terms aris-ing from the diffusion components is denoted by ,with its typical elementbeing σnm ≡σn σm ρnm .Explicit expressions for αJ n in (5)and σJ nm in (6),in termsof the parameters of the underlying returns processes,{λ,µn ,νn },are given in equations (26)and (27).In our experiment,we wish to compare the portfolio of an investor who models security returns using the pure-diffusion process in (1)with that of an investor who accounts for systemic risk by using the jump-diffusion process in (4)but matches the first two moments of returns.Thus,we need to choose the parame-ters of the jump-diffusion processes in such a way that the first two moments for this process given in equations (5)and (6)match exactly the first two moments of the pure-diffusion returns process in equations (2)and (3).7Even though it is straightforward to do this,we highlight it in a proposition because this result is important for understanding our analysis.P ROPOSITION 1:In order for the ﬁrst and second moments from the jump-diffusion process to match the corresponding moments from the pure-diffusion process,we set,for n ,m ={1,...,N },αn =ˆαn −αJ n ,(7)σnm =ˆσnm −σJ nm .(8)One interpretation of the above compensation of the parameters is that the in-vestor using the jump-diffusion returns process takes the total expected 7An important difference between our work and that of Liu et al.(2003)is that while we control for the magnitude of the first two moments when comparing the portfolio strategy that accounts for jumps with the one that ignores jumps,they do not.2814The Journal of Financereturn on the asset,ˆαn,and the covariance,ˆσnm,and subtracts from themαJ n and σJ nm,respectively,with the understanding that this will be added back through the jump term,(˜J n−1)dQ(λ).In this way,she reduces the expected return and covariance coming from the diffusion terms in order to offset exactly the contribution of the jump.Even though the unconditional expected return and covariance under the compensated jump-diffusion process will match those from the pure-diffusion process,the two processes will not lead to identical portfolios.This is because the jump also introduces skewness and kurtosis into the returns process(see equations(24)and(25)).8In the next section,we analyze the difference between the portfolio of an investor who allows for systemic jumps in returns and an investor who ignores this effect.II.Portfolio Selection in the Presence of Systemic RiskIn this section,we formulate and solve the portfolio selection problem when returns are given by the jump-diffusion process in(4).Given that financial markets are incomplete in the presence of jumps of random size,we determine the optimal portfolio weights using stochastic dynamic programming rather than the martingale pricing approach.Our modeling choices are driven by the desire to develop the simplest possible framework in which one can examine the portfolio selection problem in the presence of systemic risk.Hence,we work with a model that has a constant investment opportunity set;an extension of this model to the case where the investment opportunity set is changing over time,via shifts in the likelihood of systemic jumps,is considered in Das and Uppal(2003).Also,we model the portfolio problem in continuous time because of the analytical convenience this affords.Finally,we describe the model in the context of international portfolio selection,but the model applies to any set of securities with appropriate returns processes.A.Optimal Portfolio WeightsWe consider a U.S.investor who wishes to maximize the expected utility from terminal wealth,9W T,with utility being given by U(W T)=W1−γT,whereγ>0,1−γγ=1,so that constant relative risk aversion is equal toγ.10The investor can allocate funds across n={0,1,...,N}assets:a riskless asset denominated in U.S.dollars(n=0),a risky U.S.equity index(n=1),and risky foreign equity indexes,n={2,...,N}.8Jumps are not the only way of introducing skewness and kurtosis into the process for returns—stochastic volatility would also generate such effects.Of course,jumps have the additional effect that they constrain portfolio weights in order to prevent wealth from becoming negative.9We do not consider intermediate consumption because it has no effect on the optimal weights in our model.10For the case whereγ=1,the utility function is given by U(W T)=ln W T.Systemic Risk and International Portfolio Choice2815 The price process for the riskless asset,S0,isdS0=r S0dt,(9) where r is the instantaneous riskless rate of interest,which is assumed to be constant over time.The stochastic process for the price of each equity index(in dollar terms)11with a common jump term is as given in equation(4),which is restated below:dS nS n=αn dt+σn dz n+(˜J n−1)dQ(λ),n=1,...,N,(10)withαn andσnm defined in equations(7)and(8).Denoting the proportion of wealth invested in asset n by w n,n={1,...,N}, the investor’s problem at t can be written asV(W t,t)≡max{w n}EW1−γT1−γ,(11)subject to the dynamics of wealthd W tW t=[w R+r]dt+w (σ·dZ t)+w J t dQ(λ),W0=1,(12)where w is the N×1vector of risky-asset portfolio weights,R≡{α1−r,...,αN−r} is the excess-returns vector,σis the vector of volatilities,dZ is the vector of diffusion shocks,with the dot productσ·dZ t denoting element-by-element multiplication ofσn and dZ n,and J t≡[˜J1−1,˜J2−1,...,˜J N−1] is the vector of random jump amplitudes for the N assets at time t.The covariance matrix of the diffusion component of the joint stochastic process is given byΣ. Using the standard approach to stochastic dynamic programming and the appropriate form of Ito’s lemma for jump-diffusion processes,one can obtain the following Hamilton–Jacobi–Bellman equation0=max{w}∂V(W t,t)+∂V(W t,t)W t[w R+r]+12∂2V(W t,t)2W2t w Σw+λE[V(W t+W t w J t,t)−V(W t,t)],(13)where the terms on the first line are the standard terms when the processes for returns are continuous,and the term on the second line accounts for jumps in returns.11The dollar return on a foreign equity index includes the return on currency and the return on the international equity index in local-currency terms.For international equity returns,one could model separately the equity return in local-currency terms and the return on currency.We do not do this because it complicates the notation without adding any insights.2816The Journal of FinanceWe guess(and verify)that the solution to the value function is of the following form:V(W t,t)=A(t)W1−γt1−γ.(14)Expressing the jump term using this guess for the value function(details are in the proof for the proposition below),and simplifying the resulting differential equation,we get an equation that is independent of wealth:0=max{w}1A(t)dA(t)dt+(1−γ)[w R+r]−(1−γ)γ2w Σw+λE(1+w J t)1−γ−1.(15)Differentiating the above with respect to w,one gets the following result.P ROPOSITION2:The optimal portfolio weights in the presence of systemic risk are given by the solution to the following system of N nonlinear equations:0=R−γΣw+λEJ t(1+w J t)−γ,∀t.(16)Note that(16)gives only an implicit equation for the unconditional portfolio weights,w.Thus,to determine the magnitude of the optimal portfolio weights, one needs to solve this equation numerically,which we do in Section III.In contrast to the above solution,an investor who ignores the possibility of systemic jumps and assumes the standard model in which price processes are multivariate diffusions without jumps will choose the portfolio weights given by the familiar Merton(1971)expression below.C OROLLARY1:The weights chosen by an investor who assumes that returns are described by the pure-diffusion process in equation(1)areˆw=1Σ−1ˆR.(17)The difference between the portfolio of the investor who accounts for systemic jumps,w,and that of an investor who ignores this feature of the data and chooses portfolioˆw can be understood by comparing equation(16)and(17): The two equations are the same when there are no jumps(λ=0).Thus,the difference between w andˆw arises from the higher moments ignored in(17).12 12This also shows how our model is related to that of Chunhachinda et al.(1997),who use polynomial goal programming in a single-period model to examine the effect of skewness on portfolio choice by assuming a utility function defined over the moments of the distribution of returns.In contrast,we work with the standard power utility function that is commonly used to examine optimal portfolio selection and instead modify the returns process to allow for the possibility of skewness and higher moments.Systemic Risk and International Portfolio Choice2817B.Certainty Equivalent Cost of Ignoring Systemic RiskAbove,we have compared the optimal portfolio weights for an investor who accounts for systemic jumps in returns and the investor who ignores this feature of the data.In this section,we compare the certainty equivalent cost of following the suboptimal portfolio strategy.The objective of this exercise is to express in dollar terms the cost of ignoring systemic risk.In order to quantify the cost of ignoring systemic jumps,we compute the addi-tional wealth needed to raise the expected utility of terminal wealth under the suboptimal portfolio strategy to that under the optimal strategy.In this com-parison,we denote by CEQ the additional wealth that makes lifetime expected utility underˆw,the portfolio policy that ignores systemic risk,equal to that under the optimal policy,ing the notation V(W t,t;w i),where w i={w,ˆw} indicates the particular portfolio weights used to compute the value function,the compensating wealth,CEQ,is computed as follows:V((1+CEQ)W t,t;ˆw)=V(W t,t;w).(18) Then,from equations(14)and(18),we haveA(t;ˆw)11−γ((1+CEQ)W t)1−γ=A(t;w)11−γW1−γt,(19)which implies thatCEQ=A(t;w)A(t;ˆw)1/(1−γ)−1,(20)where,from the proof for Proposition2,A(t;w i)≡e((1−γ)[w i R+r]−12γ(1−γ)w iΣw i+λE[(1+w i J t)1−γ−1])(T−t),(21) with w i={w,ˆw}.III.Calibrating the Effect of Systemic RiskIn this section,we evaluate the effect of systemic risk on portfolio choice by calibrating the jump-diffusion model to returns on the U.S.equity index,and to five international equity indexes.This section is divided into three subsections. In the first,we describe the data and explain how we use the method of moments to estimate the parameters of the returns processes.In the second,we evalu-ate the effect of systemic risk on portfolio policies using the estimated values for the parameters of the returns process.In the third subsection,we evaluate the sensitivity of our results to the choices we have made in undertaking the calibration exercise.2818The Journal of FinanceA.Description of the Data and Estimation of the ModelThe data for the developed countries consist of the month-end U.S.dollar values of the equity indexes for the period January 1982to February 1997for the United States (U.S.),United Kingdom (U.K.),Switzerland (SW),Germany (GE),France (FR),and Japan (JP).The data for emerging economies are for the period January 1980to December 1998,and consist of the beginning-of-month value of the equity index for the United States (USA),Argentina (ARG),Hong Kong (HKG),Mexico (MEX),Singapore (SNG),and Thailand (THA).To distinguish the two sets of data,we abbreviate the countries in the developed-economy data set with two characters and denote countries in the emerging-economy data set with three characters.Table I reports the descriptive statistics for the continuously compounded monthly return on index j in U.S.dollars,r jt ,which is defined as the ratio of the log of the index value at time t and its lagged value:r j t =ln S j t j ,t −1 ,where S jt is the U.S.dollar value of the index at time t .Examining first the moments for developed economies,we observe from Panel A of Table I that the excess kurtosisTable IDescriptive Statistics for Equity Returns—UnivariatePanel A of this table gives the first four moments of the monthly returns in U.S.dollar terms for the developed-country indexes and Panel B gives the same information for the U.S.and five emerging markets.The data for the developed countries are for the period January 1982to February 1997,and include 182observations of month-end values of the equity indexes for the United States (U.S.),United Kingdom (U.K.),Japan (JP),Germany (GE),Switzerland (SW),and France (FR).The data for emerging economies consist of 227observations of the beginning-of-month value of the equity indexes for the USA,Argentina (ARG),Hong Kong (HKG),Mexico (MEX),Singapore (SNG),and Thailand (THA)for the period January 1980to December 1998.Panel A:Developed CountriesU.S.U.K.JP GE SW FR Avg.Mean0.01020.00840.00800.01200.01020.01130.0100Standard deviation0.04200.05670.06970.05760.05150.06110.0564Skewness−1.1648−0.4623−0.0508−0.2308−0.6382−0.4325−0.4966Significance level0.00000.01150.78150.20740.00040.0181Excess kurtosis7.2236 1.92120.8754 2.9546 5.5405 1.5780 3.3489Significance level 0.00000.00000.01800.00000.00000.0000Panel B:Emerging CountriesUSAARG HKG MEX SNG THA Avg.Mean0.01040.00400.00760.00340.00650.000040.0053Standard deviation0.04140.21530.10260.14370.07720.10370.1140Skewness−1.13530.1187−1.4163−2.0224−0.7684−0.6077−0.9719Significance level0.00000.46810.00000.00000.00000.0002Excess kurtosis6.1823 6.2377 6.93889.1851 4.8603 3.7800 6.1974Significance level 0.00000.00000.00000.00000.00000.0000of returns is substantially greater than that for normal distributions(in the table,we report kurtosis in excess of3,which is the kurtosis for the normal distribution).The excess kurtosis in the data ranges from0.87for France to7.22 for the U.S.For the data on emerging economies,as one would expect,the excess kurtosis is much greater,ranging from3.77for Thailand to9.18for Mexico.All 12kurtosis estimates are significant.There are two possible reasons for the kurtosis:(1)When the multivariate return series is not stationary,the mixture of distributions results in kurtosis;and(2),if the returns are characterized by large shocks,then the outliers inject kurtosis.The second feature of the data is that the skewness of returns for all the developed-market indexes is negative,and for the emerging-country indexes it is more strongly negative,except for Argentina,where it is insignificantly different from zero.The negative skewness is a well-known feature of equity index time series over this time period(1982–1997).Within this period,there were several large negative shocks to the markets contributing to the negative skewness:for instance,the market crash of October1987,the outbreak of the Gulf War in August1990,the Mexican crisis in December1994,and the Russian crisis in August1998.13Table II reports the covariances and correlations between the returns on the international equity indexes.The correlations for the developed countries range from a low of0.33between the United States and Japan,to a high of 0.68between Germany and Switzerland.The average correlation between the equity markets for developed countries is0.51.For the emerging countries,the correlations range from the very low0.05between Hong Kong and Argentina to0.55between Singapore and the United States.The average correlation for the emerging countries is0.31,which,as one would expect,is much lower than that for the developed countries.For the benchmark case of the pure-diffusion process in equation(1),the parameters to be estimated are{ˆα, Σ},with the moment conditions available being the ones in equations(2)and(3).From these moment conditions we see that{ˆα, Σ}can be estimated directly from the means and the covariances of the data series.To derive the unconditional moments of the jump-diffusion returns processes in(4),we identify the characteristic function by exploiting its relation to the Kolmogorov backward equation.14Differentiating the characteristic function then gives the moments of the returns process.The expressions for the moments of the continuously compounded returns are the following:for n,m={1,...,N},mean=tαn−12σ2n+λµn,(22)covariance=t[σnm+λ(µnµm+νnνm)],(23) 13The negative skewness arises also because volatility tends to be higher when returns are negative.14Details of this derivation are given in Das and Uppal(2003).。

电子数字钟的设计外文资料中英文翻译

外文资料TIME AND FREQUENCY MEASUREMENTSEV ALUATION OF CIRCUITS FOR SYNCHRONIZATION OF ELECTRONIC CLOCKS WITH STAY~ARD TIME AND FREQUENCY SIGNALSA Markovian mathematical model is proposed for a comparative analysis of noise suppression and locking time efficiency of different electronic clock synchronization systems with time codes transmitted by the State Time and Frequency Service. The synchronization circuit of the ChK7-50 commercial clock was found the most efficient.The invention of radio made possible the development of many modern branches of science and engineering using data transmission over radio channels including the transmission of accurate time signals to remote stations. The sources of information on current time are electronic clocks 1 (Fig. 1) with high-stability reference oscillators. All known electronic-clock synchronization circuits [1] correspond to the generalized block diagram in Fig.1. Time information X(t) is received at destination points as a sequence of time code words as a part of the standard time and frequency signal transmitted by the State Time and Frequency Service. At the receiving end of such a system the signal is demodulated and the received information is decoded before being applied to the synchronization system receiver as a word of more than 100 bits U(O (in a format containing information on hours, minutes, seconds, etc.).The well-known method of estimating the efficiency of a synchronization system by its noise suppression capability only, and its overall performance by several other criteria reflecting other characteristics, was not applied. Here we propose a method of estimating the synchronization system efficiency by criteria that consider both noise suppression and the duration of transients in the system.Simplifying Assumptions.To derive exact analytic expressions for synchronization system efficiency, the code word length is assumed to be I,i.e., we consider not its current time "contents," but its "correctness" (level 0) or "incorrectness" (level 1) only. Because of this, the standard signal X(t), the noise signal ~(t) originating along the signal propagation path, the local clock input signal U(t), and the signal of the local clock 3 X1(t) are scalars assuming the values 0 or 1. Under the above assumptions, the circuit 2 can implement different "< >" comparison operations: modulo 2 addition (¡°+ "), the OR function "v ," and other operations corresponding to the comparison of, e.g., one-bit and multibit code words U(t) and Xl(t).The result of comparison U(t) and Xt(t) is an error signal Z(t) applied to thecontrol automaton 4 with an internal state X2(t) E [(3, 1 . . . . . q-1] that generates a signal B(t) E [0, 1] that controls the switch 5.Switch 5 controls the passage of the input signal U(t) to the time set inputs of the local clock. The control automaton of the synchronization system can be either a modulo q counter or a shift register with n = log2q bits.Mathematical Model of Synchronization System.Based on the above assumption, the equations of the dynamics of the components of the combined statec(t)=(x~(t), x,(t) TX x ( l + l ) = ( l - - B ( t ) ) X l ( t ) ' k B ( l ) ~ ( t ) ;n(t) =~(x,(t)-(q-I )); ~ (1)X,(t+t)=lt-~(X~(O<>~(O)lx IX[X,(t)+ l]modq Jwhere r = 0 if X ~ 0 and r = I if X = O.Let us now consider versions corresponding to comparison of one-bit code words (version a) and multibit code words using the operations < > = "+" and < > = "v" respectively.Expression (I) takes into account the condition of "standardness" of the signal XT(t) = 0 according to which U(t) =7~(t) < > /~(t) = ~(t).The synchronization system algorithm (1) includes the following operations: at the instant the local clock is turned on, its initial reading Xt(t) = U(t) is automatically set according to the received code; after one code word interval, tile received time code is compared with the local clock code;after one code word interval, tile received time code is compared with the local clock code;if the codes match, the clock setting is assumed to be correct, no new setting takes place, and the state of the control automaton X2(t), which counts the number of consecutive mismatches, is reset to 0;if the codes do not match and the number of consecutive mismatches reaches the threshold level q, the clock readings are assumed to be incorrect and are automatically corrected in accordance with (1) to a new reading X 1 = U.Problem Statement.The problem is to fred a method of estimating the efficiency of synchronization systems fhat considers both noise suppression and the locking time, using the equation of dynamics (1) of some known system, and certain assumptions about the noise ~(t).The method discussed below consists in deriving from the equation of dynamics a transition graph of the synchronization system states together with the transitionprobabilities found from the noise characteristics, determination of the transition probability matrix, and calculation of the final probabilities and the mean time of transition of the system from its initial state to full synchronism applying the theory of finite Markov chains.State Transition Probability Graph.A state graph (Fig. 2) and a transition probability matrix are constructed from the equations of dynamics of every synchronization system, assuming that the system passes to the next state with a noise probability p = Prob (~ = 1) = Prob (U = 1) if U = ~ = I and with a noise probabilityl-p= Prob(u=o) =r-rob (~=o). (2)ifu=~ =o.The matrix of the probabilities of transition from one state to another in one step in the graph in Fig. 2 has the form-l--pp o..,o o o o .,. o -'l--p 0 p...O 0 0 0 . . . 01-p00...p o 0 0 .., o1-po o...o p o o ... oP= o o o...o ~ 1-p o ... o (3)0 0 0...0 p 0 l - - p . , . 04 "o'Gii:o;; i::iG__l--pO 0...0 p 0 0 . , . 0 __Noise Suppression Efficiency.Since the system transition probabilities are independent of time and the graph has no absorbing states, the operation of the synchronization system can be analyzed applying the methods of the theory of ergodic Markov chains [2].In [l] was introduced and analyzed the criterion of synchronization system noise suppression efficiency--- =Prob (X~(O =l)/p,where p is the probability of a code error at the synchronization system input: Prob[Xl(t) = 1] = tIq+ t + ... + II2q, and II2q are final probabilities of states q + 1 . . . . . 2q. These probabilities are found from the matrL,~ equation (E - P/) II = 0 considering the normalization condition, where E is a unit matrix, pr is the transpose matrix of the probabilities of transition from state to state in a single step, and II is a (2q - D-dimensional column vector. In a synchronization system with modulo q counting of mismatches, the criterion e of versions a and b is given respectively by~a=Pq-~[l --(l--piqlf{ ( ~--pq)(1--p)r + p#-i[ l ----(1--p)r~=p,T-~q([l+pq(q_l)l.Locking Time Efficiency.The noise suppression efficiency criterion proposed in [t] does not allow for the synchronization system transients. The transient process is analyzed by the conventional method of absorbing Markov ehahas [2]. The matrix P is replaced by i0 in which "I" is assumed to be the absorbing state. To be definite, the control automaton state and initial readings of the local clock at turn-on are set to the state "q + 1." The transient time of the transition of the synchronization system from any state, except the initial one, to the correct state "I" is given by the relation~=(e_Q)-~ ~,, where r is a (2q - l)-dimensional column vector of the time of transition from any state to state "I" (the initial state mentioned above requires one component rq+l), Q is a [(2q - l)(2q - 1)]-dimensional submatrix of the matrix P = (E l Q/R [ Q); P is the system transition matrix, with the absorbing state "1," and ~q = (I, 1 . . . . . t) t is a (2q - D-dimensional column vectorThe locking time rq+ t of a synchronization system with modulo q counting of mismatch sequences in comparison operations < > = "+" and "V" is given respectively %+1 =if -(l -p)qlflp(12-p)~l; (5a)"=~,+l=q/(l--p). (5b)Joint Efficiency Criterion.The joint efficiency criterion,9= t t(~%+~) (6)allows practical selection of the best synchronization system. For example, From Tables I and 2 of different synchronizationsystems with p -~ 0 described in [I] follows that the circuit f with a shift register as the control automaton is preferable if q= 2 and the circuit b, discussed here as an example, if q = 4.Conclusions.1. A method, based on a mathematical Markovian model, has been proposed for estimating the noise suppression and locking time efficiency of systems for synchronizing electronic clocks by standard time and frequency signals.2. It is shown that if the number of control automaton states q (q = 4) is large and the probability of errors is low (p--, 0), the preferable synchronization system is the system b with an OR comparison circuit, as used in the design of the CHK7-50 radio synchronized electronic clock.REFERENCES1. V. V, Akulov, Izmer, Tekh., No. t0, 28 (1994).2. J. Kemeny and J. Snell, Finite Markov Chains [Russian translation], Nauka, Moscow (1970).。

解析数论是使用数学分析作为工具来解决数论问题的分支

解析数论是使用数学分析作为工具来解决数论问题的分支。

微积分和复变函数论发展以后，产生了解析数论。

该学科的第一个主要成就是狄利克雷用解析方法证明了Dirichlet's theorem on arithmetic progressions。

依靠黎曼zeta函数对素数定理的证明是另一个里程碑。

解析数论是解决数论中艰深问题的重要工具，数论中有些问题必须由解析方法才能提出或解决。

中国的华罗庚、王元、陈景润等人在“哥德巴赫猜想”、“华林问题”等解析数论问题上取得世界公认的成就。

黎曼ζ函数Riemann zeta functionIn mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of great significance in number theory because of its relation to the distribution of prime numbers. It also has applications in other areas such as physics, probability theory, and applied statistics. The Riemann hypothesis, a conjecture about the distribution of the zeros of the Riemann zeta function, is considered by many mathematicians to be the most important unsolved problem in pure mathematics.[1]DefinitionThe Riemann zeta-function ζ(s) is the function of a complex variable s initially defined by the following infinite series:As a Dirichlet series with bounded coefficient sequence this series converges absolutely to an analytic function on the open half-plane of s such that Re(s) > 1 and diverges on the open half-plane of s such that Re(s) < 1. The function defined by the series on the half-plane of convergence can however be continued analytically to all complex s≠ 1. For s= 1 the series is formally identical to the harmonic series which diverges to infinity. As a result, the zeta function becomes a meromorphic function of the complex variable s, which is holomorphic in the region {s∈ C : s≠1} of the complex plane and has a simple pole at s= 1 with residue 1.Specific valuesThe values of the zeta function obtained from integral arguments are called zeta constants. The following are the most commonly used values of the Riemann zeta function.this is the harmonic series.this is employed in calculating the critical temperature for a Bose–Einstein condensate in physics, and for spin-wave physics in magnetic systems.the demonstration of this equality is known as the Basel problem. The reciprocal of this sum answers the question: What is the probability that two numbers selected at random are relatively prime? [2]this is called Apéry's constant.Stefan–Boltzmann law and Wien approximation in physics.Euler product formulaThe connection between the zeta function and prime numberswas discovered by Leonhard Euler, who proved the identitywhere, by definition, the left hand side is ζ(s) and the infiniteproduct on the right hand side extends over all prime numbers p(such expressions are called Euler products):Both sides of the Euler product formula converge for Re(s) > 1. The proof of Euler's identity uses only the formula for the geometric series and the fundamental theorem of arithmetic. Since the harmonic series, obtained when s= 1, diverges, Euler's formula implies that there are infinitely many primes. For s an integer number, the Euler product formula can be used to calculate the probability that s randomly selected integers are relatively prime. It turns out that this probability is indeed 1/ζ(s). The functional equationThe Riemann zeta function satisfies the functional equationvalid for all complex numbers s, which relates its values at points s and 1 −s. Here, Γ denotes the gamma function. This functional equation was established by Riemann in his 1859 paper On the Number of Primes Less Than a Given Magnitude and used to construct the analytic continuation in the first place. An equivalent relationship was conjectured by Euler in 1749 for the functionAccording to André Weil, Riemann seems to have been very familiar with Euler's work on the subject.[3]The functional equation given by Riemann has to be interpreted analytically if any factors in the equation have a zero or pole. For instance, when s is 2, the right side has a simple zero in the sine factor and a simple pole in the Gamma factor, which cancel out and leave a nonzero finite value. Similarly, when s is 0, the right side has a simple zero in the sine factor and a simple pole in the zeta factor, which cancel out and leave a finite nonzero value. When s is 1, the right side has a simple pole in the Gamma factor that is not cancelled out by a zero in any other factor, which is consistent with the zeta-function on the left having a simple pole at 1.There is also a symmetric version of the functional equation, given by first definingThe functional equation is then given by(Riemann defined a similar but different function which he called ξ(t).)The functional equation also gives the asymptotic limit(GergőNemes, 2007)Zeros, the critical line, and the Riemann hypothesisThe functional equation shows that the Riemann zeta function has zeros at −2, −4, ... . These are called the trivial zeros. They are trivial in the sense that their existence is relatively easy to prove, for example, from sin(πs/2) being 0 in the functional equation. The non-trivial zeros have captured far more attention because their distribution not only is far less understood but, more importantly, their study yields impressive results concerning prime numbers and related objects in number theory. It is known that any non-trivial zero lies in the open strip {s∈ C: 0 < Re(s) < 1}, which is called the critical strip. The Riemann hypothesis, considered to be one of the greatest unsolved problems in mathematics, asserts that any non-trivial zero s has Re(s) = 1/2. In the theory of the Riemann zeta function, the set{s∈ C: Re(s) = 1/2} is called the critical line. For the Riemann zeta function on the critical line, see Z-function.The location of the Riemann zeta function's zeros is of great importance in the theory of numbers. From the fact that allnon-trivial zeros lie in the critical strip one can deduce the prime number theorem. A better result[4]is that ζ(σ+ i t) ≠ 0 whenever |t| ≥ 3 andThe strongest result of this kind one can hope for is the truth of the Riemann hypothesis, which would have many profound consequences in the theory of numbers.It is known that there are infinitely many zeros on the critical line. Littlewood showed that if the sequence (γn) contains the imaginary parts of all zeros in the upper half-plane in ascending order, thenThe critical line theorem asserts that a positive percentage of the nontrivial zeros lies on the critical line.In the critical strip, the zero with smallest non-negative imaginary part is 1/2 + i14.13472514... Directly from the functional equation one sees that the non-trivial zeros are symmetric about the axis Re(s) = 1/2. Furthermore, the fact that ζ(s) = ζ(s*)* for all complex s≠ 1 (* indicating complex conjugation) implies that the zeros of the Riemann zeta function are symmetric about the real axis.The statistics of the Riemann zeta zeros are a topic of interest to mathematicians because of their connection to big problems like the Riemann hypothesis, distribution of prime numbers, etc. Through connections with random matrix theory and quantum chaos, the appeal is even broader. The fractal structure of the Riemann zeta zero distribution has been studied using rescaled range analysis.[5] The self-similarity of the zero distribution is quite remarkable, and is characterized by a large fractal dimension of 1.9. This rather large fractal dimension is found over zeros covering at least fifteen orders of magnitude, and also for the zeros of other L-functions.The properties of the Riemann zeta function in the complex plane, specifically along parallels to the imaginary axis, has also been studied, by the relation to prime numbers, in recentphysical interference experiments, by decomposing the sum into two parts with opposite phases, ψ and ψ*, which then are brought to interference. [6]For sums involving the zeta-function at integer and half-integer values, see rational zeta series.[O] ReciprocalThe reciprocal of the zeta function may be expressed as a Dirichlet series over the Möbius functionμ(n):for every complex number s with real part > 1. There are a number of similar relations involving various well-known multiplicative functions; these are given in the article on the Dirichlet series.The Riemann hypothesis is equivalent to the claim that this expression is valid when the real part of s is greater than 1/2. [O] UniversalityThe critical strip of the Riemann zeta function has the remarkable property of universality. This zeta-function universality states that there exists some location on the critical strip that approximates any holomorphic function arbitrarily well. Since holomorphic functions are very general, this property is quite remarkable.[O] Representations[O] Mellin transformThe Mellin transform of a function f(x) is defined asin the region where the integral is defined. There are various expressions for the zeta-function as a Mellin transform. If the real part of s is greater than one, we havewhere Γ denotes the Gamma function. By subtracting off the first terms of the power series expansion of 1/(exp(x) −1) around zero, we can get the zeta-function in other regions. In particular, in the critical strip we haveand when the real part of s is between −1 and 0,We can also find expressions which relate to prime numbers and the prime number theorem. If π(x) is the prime-counting function, thenfor values with We can relate this to the Mellin transform of π(x) bywhereconverges forA similar Mellin transform involves the Riemann prime-counting function J(x), which counts prime powers p n with a weight of 1/n,so that Now we haveThese expressions can be used to prove the prime number theorem by means of the inverse Mellin transform. Riemann's prime-counting function is easier to work with, and π(x) can be recovered from it by Möbius inversion.Also, from the above (specifically, the second equation in this section), we can write the zeta function in the commonly seen form:[O] Laurent seriesThe Riemann zeta function is meromorphic with a single pole of order one at s= 1. It can therefore be expanded as a Laurent series about s= 1; the series development then isThe constants γn here are called the Stieltjes constants and can be defined by the limitThe constant term γ0 is the Euler-Mascheroni constant.[O] Rising factorialAnother series development valid for the entire complex plane iswhere is the rising factorial This can be used recursively to extend the Dirichlet series definition to all complex numbers.The Riemann zeta function also appears in a form similar to the Mellin transform in an integral over the Gauss-Kuzmin-Wirsing operator acting on x s−1; that context gives rise to a series expansion in terms of the falling factorial.[O] Hadamard productOn the basis of Weierstrass's factorization theorem, Hadamard gave the infinite product expansionwhere the product is over the non-trivial zeros ρ of ζ and the letter γ again denotes the Euler-Mascheroni constant. A simpler infinite product expansion isThis form clearly displays the simple pole at s = 1, the trivial zeros at −2, −4, ... due to the gamma function term in the denominator, and the non-trivial zeros at s = ρ.[O] Globally convergent seriesA globally convergent series for the zeta function, valid for all complex numbers s except s = 1, was conjectured by Konrad Knopp and proved by Helmut Hasse in 1930:The series only appeared in an Appendix to Hasse's paper, and did not become generally known until it was rediscovered more than 60 years later (see Sondow, 1994).Peter Borwein has shown a very rapidly convergent series suitable for high precision numerical calculations. The algorithm, making use of Chebyshev polynomials, is described in the article on the Dirichlet eta function.[O] ApplicationsAlthough mathematicians regard the Riemann zeta function as being primarily relevant to the "purest" of mathematical disciplines, number theory, it also occurs in applied statistics (see Zipf's law and Zipf-Mandelbrot law), physics, and the mathematical theory of musical tuning.During several physics-related calculations, one must evaluate the sum of the positive integers; paradoxically, on physical grounds one expects a finite answer. When this situation arises, there is typically a rigorous approach involving much in-depth analysis, as well as a "short-cut" solution relying on the Riemann zeta-function. The argument goes as follows: we wish to evaluate the sum 1 + 2 + 3 + 4 + · · ·, but we can rewrite it as a sum of reciprocals:The sum S appears to take the form of However, −1 lies outside of the domain for which the Dirichlet series for thezeta-function converges. However, a divergent series of positive terms such as this one can sometimes be represented in a reasonable way by the method of Ramanujan summation (see Hardy, Divergent Series.) Ramanujan summation involves an application of the Euler–Maclaurin summation formula, and when applied to the zeta-function, it extends its definition to the whole complex plane. In particularwhere the notation indicates Ramanujan summation.[7]For even powers we have:and for odd powers we have a relation with the Bernoulli numbers:Zeta function regularization is used as one possible means of regularization of divergent series in quantum field theory. In one notable example, the Riemann zeta-function shows up explicitly in the calculation of the Casimir effect.[O] GeneralizationsThere are a number of related zeta functions that can be considered to be generalizations of Riemann's zeta-function. These include the Hurwitz zeta functionwhich coincides with Riemann's zeta-function when q = 1 (note that the lower limit of summation in the Hurwitz zeta function is 0, not 1), the Dirichlet L-functions and the Dedekind zeta-function. For other related functions see the articles Zeta function andL-function.The polylogarithm is given bywhich coincides with Riemann's zeta-function when z = 1.The Lerch transcendent is given bywhich coincides with Riemann's zeta-function when z = 1 and q = 1 (note that the lower limit of summation in the Lerch transcendent is 0, not 1).The Clausen function that can be chosen as the real orimaginary part ofThe multiple zeta functions are defined byOne can analytically continue these functions to then-dimensional complex space. The special values of these functions are called multiple zeta values by number theorists and have been connected to many different branches in mathematics and physics.[O] Zeta-functions in fiction。

非线性混沌现象及其在激光原子相互作用和玻色爱因斯坦凝聚中的表

•Laser beam
•What happens if we replace the ultra-cold atom with the BEC?
非线性混沌现象及其在激光原子相互作用和玻色爱因斯坦凝聚中的表
2021/1/4
非线性混沌现象及其在激光原子相互作用和玻色爱因斯坦凝聚中的表
•概要 •1、非线性动力学混沌现象简介 • 保守系统的混沌 • 耗散系统的混沌 • 量子混沌 •2、应用于强激光的原子电离 •3、应用于波色---爱因斯坦凝聚体 •4、应用于纳米材料（如有时间）
非线性混沌现象及其在激光原子相互作用和玻色爱因斯坦凝聚中的表
一、非线性系统的混沌现象
• 保守系统与耗散系统区别点
相体积收缩与否
相同点
•混沌轨道有李雅谱诺夫指数描述
非线性混沌现象及其在激光原子相互作用和玻色爱因斯坦凝聚中的表
•保守系统的标准映象
非线性混沌现象及其在激光原子相互作用和玻色爱因斯坦凝聚中的表
hing
•Chirped pulse amplification
非线性混沌现象及其在激光原子相互作用和玻色爱因斯坦凝聚中的表
强激光场中的原子----引言
• 近些年来，强场与原子相互作用问题引起人们的广泛关注。这主要由于在过去几十年里激光技术和实验技术的飞速发展使得①电磁场强度与库仑场(原子玻尔半径处)在同．一量级(甚至超过)；②可以非常精确地测量光电子的能量及角分布等物理量。
作用和玻色爱因斯坦凝聚中的表
What is BECs good for
• Too new and we know too little • Potential application: Sensitive
measurement, tiny instrument, atom laser, quantum information, etc.

Expressions of thesis英语论文写作关键词,套话

Expressions to introduce the Subject:This paper reports onRecords of their noticing were tracked throughout the year and recordings of their oral output made over the same period were analyzed to determine whether there was any development in the use of the forms that the students had noticed.The paper describes an initial analysis of the tracking of two students’ noticing and subsequent use of a non-count noun, which presented them with difficulties at the start of the year.The paper considers how th e students’ noticing of the word might have related to this improvement.This study considersPresumptionI anticipated that the process of both researching and presenting on a research topic in English would help develop their L2 proficiency.I hoped thatCitationSchmidt and Frota describe how learners might notice input ‘in the normal sense of the word, that is, consciously’ (1986: 311).Schmidt has refined his concept of noticing to account for interlanguagedevelopment that is not necessarily conscious but does require attention. For Schmidt, intake is simply ‘that part of the input that the learner notices’ (1990: 139).Much of his early data came from his own efforts at learning Portuguese (Schmidt and Frota, 1986).Even so, Schmidt makes strong claims for noticing (1990: 144): ‘Those who notice most learn most, and it may be that those who notice most are those who pay attention most, as a general disposition or on particular occasions.’Swain (2000) has argued thatSwain’s output hypothesis (1995) proposes that, for the purposes of the development of syntax and morphology, it is more beneficial for learners to use the L2 actively rather than to listen to it.The second function Swain attributes to output, analogous to Gass’s idea that output feeds back into the acquisition process at the level of intake, is the chance it gives learners to test their hypotheses about the L2.Though the discourse initiation appears to be predominantly in the hands of the teacher, it looks as if, given the chance, the informants benefit more from topics initiated by the learners.(Slimani, 1989: 227)As to why student-initiated topicalization had a better chance of being noticed, Slimani suggests thatThis idea of student autonomy is relevant to natural order hypotheses in SLA (Pienemann, 1989), and a learner’s noticing of a form can be interpreted as a readiness to acquire it (e.g. Williams and Evans, 1998; Williams, 1999).Williams concludes that‘teachers cannot expect learners to consistently ferret out and notice morphosyntactic features’ (1999, p. 620).。

EFFECTS OF SHADOWING IN DOUBLE POMERON EXCHANGE PROCESSES

a r X i v :h e p -p h /9503394v 1 21 M a r 1995February 1995TAUP 2235-95CBPF FN -009/95Eﬀects of Shadowing inDouble Pomeron Exchange ProcessesE.Gotsman 1)E.M.Levin 2),3),a )U.Maor 1),3)1)School of Physics and AstronomyRaymond and Beverly Sackler Faculty of Exact SciencesTel Aviv University,Tel Aviv 69978,Israel2)Mortimer and Raymond Sackler Institute of Advanced Studies School of Physics and Astronomy,Tel Aviv UniversityTel Aviv,69978,Israel3)LAFEX,Centro Brasileiro de Pesquisas F´ısicas /CNPqRua Dr.Xavier Sigaud 150,22290-180,Rio de Janiero,RJ,BrasilAbstract:The eﬀects of shadowing in double Pomeron exchange processes are in-vestigated within an eikonal approach with a Gaussian input.Damping factors due to screening are calculated for this process and compared with the factors obtained for total,elastic and single diﬀraction cross sections.Our main conclusion is that counting rate cal-culations,of various double Pomeron exchange processes (without screening corrections)such as heavy quark and Higgs production are reduced by a factor of 5in the LHC energy range,when screening corrections are applied.The process of double Pomeron exchange(DPE),shown in Fig.1,has been recognized for sometime[1,2]as an interesting window through which we can further persue our study of Pomeron dynamics,and extend our knowledge of diﬀraction.Even though DPE pro-cesses have relatively small cross sections,they have a very clean experimental signature, where the central diﬀractive cluster is seperated from the remnants of the two projectiles by large rapidity gaps.(For a schematic lego plot see Fig.2).DPE processes have recently attracted a considerable amount of attention as a possible background for rare electroweak events[3],as well as actual sources for central diﬀraction of q¯q jets[4]and minijets[5], heavyﬂavor production[6],Higgs production[7]and an interesting conﬁguration for the study of the Pomeron structure[8].In this paper we wish to examine the consequences of including screening corrections in the initial state of DPE diagrams.Our calculations are applicable to DPE calculated either in the conventional Regge formalism,or through a two gluon exchange approxi-mation[9,10].We show that these s-channel unitarity corrections,cause DPE processes to be strongly suppressed throughout the Tevatron energy range,and even more so at higher energies.The degree of this suppression can easily be assessed in terms of a damp-ing factor<|D|2>.In this paper we proceed to calculate the damping factor for DPE processes,and make a realistic evaluation of some of the associatedﬁnal states.As we shall show,our estimates are considerably smaller than the uncorrected rates,published previously.The present investigation extends our ongoing study[11]on the implementation of s-channel unitarity corrections to Pomeron exchange in high energy hadron scattering. Our study has mostly concentrated on a supercritical DL soft Pomeron[12]α(t)=1+∆+α′twith∆≃0.085andα′≃0.25GeV−2.This simple model is rather successful in reproduc-ing the available hadronic data on total and elastic cross sections.We note,nevetheless, that our method is equally eﬀective if we choose to calculate with the hard BFKL QCD-Pomeron[13].This will be the main subject of a paper to be published shortly.In previous publications[11]we have attempted a systematic study of s-channel uni-tarity screening corrections in an eikonal approximation,where our b-space amplitude is written asa(s,b)=i(1−e−Ω(s,b))(1) To obtain analytic expressions for the cross sections of interest,we make the following simplifying assumptions:1)The opacityΩ(s,b)is a real function,i.e.a(s,b)is pure imaginary,when necessary analyticity and crossing symmetry can be easily restored[11].12)We assume our input opacity to be a GaussianΩ(s,b)=ν(s)e−b22πR2(s)(ss0](4)andσ0=σ(s0).3)Our eikonal approximation does not include diﬀractive rescattering.Neglecting these states is a reasonable approximation,asσdif fσin tot =1−∞n=1(−1)n+1νnσin el =1−4∞n=1(−1)n+1νn[2n+1−1]d2b a i(s,b)(7)where a i(s,b)is the b-space amplitude of interest,and P(s,b)=e−2Ω(s,b)denotes the probability[11]that no inelastic interaction takes place at impact parameter b.We note that the deﬁnition of<|D2i|>is correlated to the deﬁnition of the survival probability <|S|2>in the case of hard parton scattering[3].We would like to mention that the physical meaning of the damping factor in the parton approach,is the probability to have one parton shower collision in the hadron-hadron interaction.Bjorken’s survival probability is the damping factor multiplied by2the ratio of the input cross section to the inclusive one,with the same trigger in the one parton shower collision.For example the Bjorken survival probability for high p T jet central diﬀraction is<|S|2>=<|D|2>·σone parton shower(high p T jet in double pomeron collision)dM2)outdM2)in=a(1(a+n)n!(9)where4¯R2(ss0)≤2(10)and¯R2(sM2)(11) r0≪R20denotes the radius of the triple vertex and can be neglected.γ(a,2ν)= 2ν0z a−1e−z dz,denotes the incomplete Euler gamma function.One has to integrate over M2to obtain the integrated SD cross sectionσSD.Weﬁnd that a(s,M2) has a rather weak dependence on M2as it is proportional toα′lnM2(withα′small)over a relatively narrow domain.So in practice,one can factor out<|D SD|2>in the M2 integration.Thus we haveσoutSDγp→ψX,for which a(s)→2precociously,exhibit a very tempered energy dependence, which is the result of the early saturation of the screening corrections.This behaviour is to be contrasted with the eﬀective power behaviour of the total and elastic cross sections at these energies.We now turn to a detailed calculation of central diﬀraction(CD),which proceeds through DPE.This process was originally calculated by Streng[2].We follow this calcu-lation and then proceed to calculate<|D CD|2>.The relevant kinematics are shown in Fig.3.We remind the reader that the t-space elastic scattering amplitude is given byF(s,t)=14.Adopting Streng’s notation[2]we haveF(s,t)=2g2P (0)(sR2(s)( sds1ds2dt1dt2=4π2σP P(M2)s1,(qs1,(q s2,(qs2,(qds1ds2dt1dt2=4π2σP P(M2)4)4¯R4(ss2)¯ν2(ss2)·e−¯B(s2)e−¯B(s2)(18)where¯B(s2R2+α′ln(s si)s i )=σ0s i)(sNote that in Eq.(18)we have a factor4in the denominator.This is in accord with the Reggeon calculus rules for identical particles.After integrating over k21and k22we haves1s2d2σ¯R2(ss2)·g4P(0)(s2(R20+α′ln s s1we can integrate over s1and obtainM2dσ2α′σP P(M2)g40(s2(R20+α′ln sR20+α′ln sM2dM218πα′σP P(M2)g40(sR20+α′ln sM22πd2qe−i(q.b)e−1M2)q2(25)which can be rewritten asΩP P(s,b)=1M2)M2)e−2b2M2)(26)The screened expression for M2dσdM2=1M2)M2) d2be−Ω(s,b)e−2b2M2)(27)As we have already seen[11]this integral can be evaluated analytically yieldingM2dσ8πα′σP P(M2)ν2(sM2)2ν(s))aγ(a,2ν(s))(28)5where a is deﬁned asR2(s)a(s,M2)=2)≥2(29)M2This allows us to conclude that<|D CD|2>=a(1References[1]D.M.Chew and G.F.Chew,Phys.Lett.53B(1974)191.[2]K.H.Streng,Phys.Lett.166B(1986)443.[3]J.D.Bjorken,Phys.Rev.D45(1992)4077;D47(1992)101.[4]Jon Pumplin,preprint MSUHEP-41222(1994).[5]A.Donnachie and ndshoﬀ,Phys.Lett.B332(1994)433.[6]A.Bialas and W.Szeremeta,Phys.Lett.B296(1992)191;A.Bialas and R.Janik,Z.Phys.C62(1994)487.[7]A.Schafer,O.Nachtman and R.Shopf,Phys.Lett.B249(1990)331;A.Bialas andndshoﬀ,Phys.Lett.B256(1991)540.[8]G.Ingelman and P.Schlein,Phys.Lett.B152(1985)256.[9]S.Nussinov,Phys.Rev.Lett.34,(1973)1268;F.E.Low,Phys.Rev.D12,(1975)163.[10]A.Donnachie and ndshoﬀ,Nucl.Phys.B244(1984)322;(1984)189;B276(1986)690.[11]E.Gotsman,E.M.Levin and U.Maor,Phys.Lett.B309,(1993)109;Z.Phys.C57(1993)667;Phys.Rev.D49(1994)R4321;Phys.Lett.B347,(1995)424.[12]A.Donnachie and ndshoﬀ,Nucl.Phys.B231(1989)189;Phys.Lett.B296(1992)227.[13]L.N.Lipatov,Sov.Phys.JETP63,(1986)904;E.A.Kuraev,L.N.Lipatov and V.S.Fadin,Sov.Phys.JETP45,(1977)199;Ya.Ya.Balitsky and L.N.Lipatov,Sov.J.Nuicl.Phys.28(1978)822.7Figure CaptionsFig.1:The process of Double Pomeron Exchange(DPE).Fig.2:A typical lego plot for DPE.Fig.3:Relevant kinematics of our DPE calculation.Fig.4:A mapping of damping factor as a function of a(s,M2)andν(s).Fig.5:DPE contribution to the cross section for heavy quark pair production versus the CM energy of the collision.Dashed lines denote the results of the uncorrected cal-culation of Bialas and Szeremeta[6].Full lines are the results after screening corrections have been made.Table CaptionsTable I.Behaviour of the asymptotic cross section for uncorrected supercritical Pomeron model and the GLM model.Table II.Damping factors at some typical accelarator energies8Table I.GLMs∆s0s2∆s0s0s2∆s0s0s2∆s0s01σtot ln sσdiff s∆s0s0s GeV<|D tot|2><|D SD|2><|D CD(m2χ(3415))|2>0.760.6260.3445400.8200.4020.3280.940.5660.3091.000.5480.2961.170.5010.24314000.7650.2890.2231.310.4670.208This figure "fig1-1.png" is available in "png" format from: /ps/hep-ph/9503394v1This figure "fig2-1.png" is available in "png" format from: /ps/hep-ph/9503394v1This figure "fig1-2.png" is available in "png" format from: /ps/hep-ph/9503394v1This figure "fig2-2.png" is available in "png" format from: /ps/hep-ph/9503394v1This figure "fig1-3.png" is available in "png" format from: /ps/hep-ph/9503394v1。

热能与动力工程专业英语译文-第一章译文

Through the application of thermodynamic principles, modern heat engines have been developed.We are facing the reality that fossil fuel reserves are diminishing and will be insufficient in the forseeable future.Consequently, to those who study thermodynamics, increasing efficiency in the use of fossil fuels and the development of alternate sources of thermal energy are the real challenges to technology for today and tomorrow.Thermodynamics is a branch of science which deals with energy, its conversion from one form to another, and the movement of energy from one location to another. Thermodynamics is involved with energy exchanges and the associated changes in the properties of the working fluid or substance.Although thermodynamics deals with systems in motion, it does not concern itself with the speed at which such processes or energy exchanges occur.Thermodynamics, like other physical sciences, is based on observation of nature. Engineering thermodynamics consists of several parts, such as basic laws, thermal properties of the working fluids, process and cycle and so on.Energy is a primitive (原始的）property. We postulate（假定）that it is something that all matter has.Kinetic energy and potential energy are two forms of mechanical energy.A change of the total energy is equal to the rate of work done on the system plus the heat transfer to the system.Enthalpy can be used either as an extensive property H or as an intensive property h.The two terms v2/2 and gz represents kinetic energy and potential energy respectively. Although the net heat supplied to a thermodynamic system is equal to the net work done by the system, the gross energy supplied to the system must be greater than the net work done by the system.Not all of the input heat is available for producing output work because some heat must always be rejected by the system.Related to the second law statements are the concepts of availability of energy, entropy, process reversibility and thermal efficiency.In all reversible processes there is no change in the availability of the energy evolved in the process.Due to this concept of availability of energy, the following statements can be made: Only a portion of heat energy may be converted into work.Entropy S is an abstract thermodynamic property of a substance that can be evaluated only by calculation.From the above expression one can find that the value of entropy of the system will increase when the heat is transferred into the system.Processes that return to their initial state are called cyclic processes.The Carnot cycle is most efficient cycle possible operating between two given temperature levels.In the ideal Rankine cycle the efficiency may be increased by the use of a reheater section. The process of reheating in general raises the average temperature at which heat is supplied to the cycle, thus raising the theoretical efficiency.After partial expansion the steam is withdrawn from the turbine and reheated at constant pressure. Then it is returned to the turbine for further expansion to the exhaust pressure. For the portion of the heat-addition process from the subcooled liquid to saturated liquid, the average temperature is much below the temperature of the vaporization and superheating process.From the viewpoint of the second law, the cycle efficiency is greatly reduced.If this relatively low-temperature heat-addition process could be raised, the efficiency of the cycle would more nearly approach that of the Carnot cycle.The refrigeration cycle is used to transfer energy (heat) from a cold chamber, which is at a temperature lower than its surroundings.The basic refrigeration cycle consists of a sequence of processes utilizing a working fluid, called the refrigerant, usually in continuous circulation within a closed system.The refrigerant receives energy in the evaporator (cold chamber) at a temperature below that of the surroundings, and then rejects this energy in the condenser (hot chamber) prior to returning to its initial state.In the absence of friction these mechanical energies are completely interchangeable; that is, one unit of potential energy can be ideally converted into one unit of kinetic energy, and vice versa.It represents energy modes on the microscopic level, such as energy associated with nuclear spin, molecular binding, magnetic-dipole moment, molecular translation, molecular rotation, molecular vibration, and so on.In a static fluid, there is no motion of one layer of fluid relative to an adjacent layer, so there are no viscous shear forces.A knowledge of fluid statics is necessary for the solution of many familiar problems, such as the determination of total water force on a dam, the calculation of pressure variation throughout the atmosphere.With no relative motion between fluid particles, there are no shear forces acting on the element, only normal forces (due to pressure) and the gravity force.In order to solve problems in fluid flow, it is often necessary to determine the variation of pressure with velocity from point to point throughout the flow field.As one knows, a streamline is a continuous line drawn in the direction of the velocity vector at each point in the flow.For one-dimensional flow, the flow properties of which do not vary in the direction normal to the streamline, the constant in the Bernoulli equation is the same for all streamlines. The term pv is called flow work (energy/mass), the term v2/2 is the kinetic energy per unit mass; and gz is the potential energy per unit mass.There are two basic types of flow, each possessing fundamentally different characteristics. The first type is called laminar flow, the second turbulent flow.The transverse movement of a particle of fluid from a faster-moving layer to aslower-moving layer will have the effect of increasing the velocity in the slower-moving layer.The inlet length required to attain fully developed flow is dependent on the type of flow.In an analysis of flow through a pipe, we are interested in the type of flow, whether laminar or turbulent, since the shear stress and resultant frictional forces acting on the fluid vary greatly for the two types.Another way of looking at the difference between laminar and turbulent flows is to consider what happens when a small disturbance is introduced into a flow.The thickness of the laminar sublayer depends on the degree of turbulence of the main stream—the more turbulent the flow, the thinner the sublayer.We know that when a fluid flows through a pipe, the layer of fluid at the wall has zero velocity; layers of fluid at progressively greater distances from the pipe surface have higher velocities, with the maximum velocity occurring at the pipe centerline.However, even though the velocity fluctuations are small, they have a great effect on the flow characteristics.Furthermore, with the large number of random particle fluctuations present in a turbulent flow, there is a tendency toward mixing of the fluid and a more uniform velocity profile. When smoke leaves a cigarette, it travels upward initially in a smooth, regular pattern; at a certain distance above the cigarette, however, the smoke breaks down into an irregular pattern.Even in turbulent pipe flow, with the great majority of the flow characterized by rough, irregular motions, there will always be a thin layer of smooth laminar flow near a wall, for the particle fluctuations die out near a boundary.When the central of core region of the flow disappears, the flow is termed fully developed viscous flow.The science of heat transfer is concerned with the analysis of the rate of heat transfer taking place in a system. Heat flow will take place whenever there is a temperature gradient.Heat conduction is the term applied to the mechanism of internal energy exchange from one body to another, or from one part of a body to another.Heat conduction is realized by the exchange of the kinetic energy of the molecules by direct contact or by the drift of free electrons in the case of heat conduction in metals.The Fourier law may be used to develop an equation describing the distribution of the temperature throughout a heat-conducting solid.The term “steady state conduction” was defined as the condition which prevails when the temperatures of fixed points within a heat-conducting body do not change with time.The te rm “one-dimensional” is applied to a heat conduction problem when only one space coordinate is required to describe the distribution of temperature within a heat-conducting body.The solution of heat conduction problems involves, in general, the writing of the general heat conduction equation in terms of the appropriate number of arbitrary constants and then the evaluating of these constants by use of the imposed boundary conditions.The electrical analogy may be used to solve more complex problems involving both series and parallel thermal resistances.When fluid flows over a solid body or inside a channel while temperatures of the fluid and the solid surface different, heat transfer between the fluid and the solid surface takes place as a consequence of the motion of fluid relative to the surface.The multiplicity of independent variables results from the fact that convection transfer is determined by the boundary layers that develops on the surface.The velocity boundary layer is defined as the thin layer near the wall in which one assumes that viscous effects are important.It should be emphasized that a thermal boundary layer can also be defined as the region between the surface and the point at which the fluid temperature has reached a certain percentage of the fluid temperature.The thermal boundary layer is generally not coincident with the velocity boundary layer, although it is certainly dependent on it.Numerous analytic expressions are available for the prediction of heat transfer coefficient in laminar tube flow.There are numerous important engineering applications in which heat transfer for flow over bodies such as a flat plate, a sphere, a circular tube, or a tube bundle are needed.The temperature variation within the fluid will generate a density gradient which, in a gravitational field, will give rise, in turn, to a convective motion as a result of buoyancy forces.The fluid motion set up as a result of the buoyancy force（浮力）is called free convection, or natural convection.The flow velocity in free convection is much smaller than that encountered in forced convection; therefore, heat transfer by free convection is much smaller than that by forced convection.According to the different condensing situation, condensation can be divided into filmwise condensation and dropwise condensation.The phenomenon of heat transfer in boiling is extremely complicated because of a large number of variables involved and very complex hydrodynamic developments occurring in the process.All bodies continuously emit energy because of their temperature, and the energy thus emitted is called thermal radiation.The radiation energy emitted by a body is transmitted in the space in the form of electromagnetic waves according to Maxwell’s classic electro magnetic wave theory or in the form of discrete photons according to Planck’s hypothesis(假说）.The emission or absorption of radiation energy by a body is a bulk process; that is, radiation originating from the interior of the body is emitted through the surface.Heat exchangers are devices that facilitate heat transfer between two or more fluids at different temperatures.The C.O.P. of a refrigerating machine is ratio of Refrigerating effect to Work input.The C.O.P. of a refrigerator, unlike the efficiency of a heat engine can be much larger than unity.The essential parts of a vapor compression system are Evaporator Compressor condenser, and Expansion valve.There are three types of vapor compressor: reciprocating, rotary, centrifugal.A vapor absorption system uses heat (thermal) energy to produce refrigeration.In an absorption system, the commonly used working substance is a solution of refrigerant and solvent.The four important factors involved in a complete air conditioning installation are:(i) Temperature control, (ii) Humidity control , (iii) Air movement and circulation, (iv) Air filtering, cleaning and purification .Give some applications of refrigerationdomestic refrigerationcommercial refrigerationindustrial refrigerationManufacture and preservation of medicinesPreservation of blood and human tissuesProduction of rocket fuelsComputer functioningmarine and transportation refrigerationWhat is a vapor compression system?A typical Vapor Compression Refrigeration SystemComponentsEvaporator: Heat exchangers for refrigerant to absorb heat from refrigerated space Compressor: to raise the temperature and pressure of refrigerant by compression Condenser: Heat exchangers for refrigerant to reject heat to the environmentReceiver tank: a reservoir to store the liquid refrigerantExpansion valve: or Refrigerant flow control, to reduce refrigerant pressureCycle diagramsWhat is the working principle of a vapor absorption system?Absorption refrigeration cycleA vapor absorption system uses heat (thermal) energy to produce refrigeration.In an absorption system, the commonly used working substance is a solution of refrigerant and solvent, such as Ammonia/water and Water/lithium bromide.A absorption refrigeration system also contains an evaporator and condenser which operate in exactly same way as for vapor compressor cycle.There is a second circuit around which an absorbent or solvent fluid flows. The evaporated refrigerant vapor is absorbed into the solvent at low pressure, and there is a net surfeit of heat for this process.The solvent, now diluted by refrigerant is raised to the high pressure by a liquid pump. High pressure refrigerant vapor is then produced by the addition of heat to the mixture, in the generator.Nuclear energy results from changes in the nucleus of atoms.As a nucleus splits, it releases a tremendous amount of heat.The nucleus splitting process is completely fissioned, it will create as much heat as the burning of 1500 short tons of coal.In 1911 the physicist Ernest Rutherford first discovered the existence of a subatomic particle, later referred to as the nucleus.In 1938, two German chemists, Otto Hahn and Fritz Strassmann reported they had produced the element barium by bombarding(轰击) uranium with neutrons.This reaction had in fact split an uranium nucleus into two nearly equal fragments(碎片）, one of which was a barium（钡）nucleus and another was a krypton（氪）nucleus.Albert Einstein developed his famous relativity theory and related the matter to energy by the equation E=mc2.Cadmium（镉）rods were used to control the chain reaction.By 1960, nuclear power generating systems in the range of 150 to 200 MW were in commercial operation.Free neutron capture upsets the internal force, which holds together the tiny particles called protons and neutrons in the nucleus.Besides the heat energy produced, fission releases an average of two or three neutrons and such nuclear radiation as gamma rays.If one of the neutrons emitted is captured by another fissionable nucleus, a second fission takes place in the manner similar to the first.When the fission becomes self-sustaining, the process is called a chain reaction.Nuclear reactors used for electric power generation consist of four main parts.They are (1) the fuel core, (2) the moderator and coolant, (3) the control rods, and (4) the reactor vessel .The fuel core contains the nuclear fuel and is the part of the reactor in which the fission takes place.In fission process the fertile materials( for instance, the U-238 ) are converted to fissile. Fertile: 可变成裂变物质的The nuclear fuel is generally contained in cylindrical rods surrounded by cladding materials，such as aluminum(铝), magnesium(镁), zirconium(锌), stainless steel, and graphite(石墨). The moderator is the substance used in nuclear reactor to reduce the energy of fast neutrons to thermal neutrons.The reactor coolant is used to remove heat from the reactor fuel core, including light water, heavy water, air, carbon dioxide, helium, sodium(钠), potassium(钾), and some organic liquids.Control rods are long metal rods that contain such elements as boron硼, cadmium镉, or hafnium罕. These elements absorb fast neutrons and therefore help control a chain reaction.The reactor vessel is a tanklike structure that holds the reactor core and other internals. The two principal types are the PWR and BWR. Both reactors use enriched uranium and light water as coolant as well as moderator.The coolant first flows downward through the annular space between the shield wall and the core barrel into a plenum at the bottom of the vessel.Then the coolant reverses its direction and flows upward through the fuel core.The heated coolant is collected at the upper plenum and exits the vessel through outlet nozzles.A reactor coolant system is usually designed with two or more closed coolant loops connected to the PWR, each containing its own steam generator and coolant pump.The steam-water mixture from the tube bundle passes through a steam swirl旋转vane叶片assembly where steam is separated from the water.In addition to the steam generator, each coolant loop in the PWR has its own pump.An electrically heated pressurizer is connected to one of the coolant loops and is used to serve the whole coolant system.The pressurizer is used to maintain the coolant pressure during steady-state operation, and to limit the pressure changes, preventing the pressure from exceeding the design limit. Boiling water nuclear steam supply system mainly consists of reactor vessel and reactor coolant circuits.Unlike the PWR, BWR system does not have the intermediate heat exchanger, or steam generator, between the coolant loop and the feedwater and steam system.For a BWR plant, steam is generated within the nuclear reactor and transferred directly to the steam turbine.A disadvantage of the BWR system is that radioactive carry-over into the steam must be guarded against and special provisions made to reduce leakage at the shaft seals of the turbine.The plant, having a peak capacity of 12 kWe, has been intended as a demonstration and a pilot plant for electricity production from solar energy.The plant is composed of three main parts: a field of flat plate solar water collectors (primary circuit); a hot water storage tank (interface); and a turbo-generator group (secondary circuit).The operating mechanism of the plant is based on the principle of converting solar energy into thermal energy.The converted energy is then stored in the hot water storage tank until reaching a temperature level of 100°C (called the index temperature), which triggers the startup of the turbo-generator group operation.The primary circuit of the plant consists of 396 flat plate solar collectors covering a net aperture area of 760 m2.The converted solar energy into thermal energy is stored in a sensible heat form within a water storage tank.The geometry of the storage tank presents the advantage of favoring the forming of a thermal stratification within the storage.The turbo-generator group (TGG) consists of an evaporator, a turbine, a condenser and an alternator.The evaporator and the condenser are both heat exchangers made of copper tubes allowing the heat transfer between the fluid and both the hot and cold sources.The turbine is of a single stage type characterized by an axial flux having a rotation speed of 900 rpm.A parabolic concentrator unit is designed to increase the temperature at the bottom of the storage tank whenever the climatic conditions are favorable.。

The Specific Heat of a Trapped Fermi Gas an Analytical Approach

C kB
=
−
1 2
x2
∂u ∂x
,
N,w
(8)
where kB is Boltzmann’s constant. The simplest approximation consists in replacing the triple sums in (6) by a triple inte-
gral. Thus,
and Rokhsar in that we provide an analytical expression for the speciﬁc heat of a non-self-
interacts in the context of a semi-classical approximation. We
C=
∂U ∂T
=1 w N,w 2
∂u ∂T
.
N,w
(7)
Note that the partial derivative at constant particle number and trap frequency is the natural way of deﬁning the speciﬁc heat for a gas in the situation concerned. After a change of variables from T to x this becomes
w1 = w ,
w2 = λw , w3 = λ′w ,
x=β w,
µ= w
1
+
λ+ 2
λ′
−
ǫ
,
(4)
where x, ǫ, λ and λ′ are newly deﬁned dimensionless variables. The internal energy becomes

数学专业英语25

However, our primary purpose is to discuss calculus. Concepts from analytic geometry that are required for this purpose will be discussed as needed.
不过，我们的主要目的是讨论微积分。为此，解析几何中的概念将只是在必要时讨论。
R.笛卡儿提出了一种好得多的办法，并建立了解析几何（也称为笛卡儿几何）这个学科。
Descartes’ idea was to represent geometric points by numbers. The procedure for points in a plane is this :
笛卡儿的思想就是用数来表示几何点，在平面上找点的过程如下：
如果想拓展微积分的范围与应用，需要进一步研究解析几何，而这种研究需用到向量的方法来实现。
Until then, all that is required from analytic geometry is a little familiarity with drawing graph of function.
一个几何图形是满足一个或多个特殊条件的点集，比如平面上的曲线。
By translating these conditions into expressions, involving the coordinates x and y, we obtain one or more equations which characterize the figure in question.
A deeper study of analytic geometry is needed to extend the scope and applications of calculus, and this study will be carried out using vector method.

坝底水浮力对重力坝稳定性的影响分析

第27卷第8期岩石力学与工程学报V ol.27 No.8 2008年8月Chinese Journal of Rock Mechanics and Engineering Aug.，2008INFLUENCES OF HYDRAULIC UPLIFT PRESSURES ON STABILITY OFGRA VITY DAMUTILI Stefano1，2，YIN Zhenyu 2，JIANG Mingjing3(1. Department of Civil Engineering，Politecnico di Milano，Milan20133，Italy；2. Department of Civil Engineering，University of Strathclyde，Glasgow G4ONG，United Kingdom；3. Department of Geotechnical Engineering，Tongji University，Shanghai200092，China)Abstract：A study of the influences of the hydraulic uplift pressures underneath the base of a typical concrete gravity dam on its stability is presented. The dam is located at Cumbidanovu(Sardegna，Italy). The foundation of the dam is made of heavily fractured rock. Firstly，analytical calculations about the equilibrium of the dam as a free body have been employed to evaluate the maximum hydraulic pressure before collapsing and to assess the impact of an effective drainage system on the stability of the dam in a simple way. Secondly，numerical analyses by the distinct element method(DEM) using the code UDEC have been carried out to evaluate the hydraulic flow taking place within the fractured rock foundation，the uplift pressure distribution generated by the calculated flow，and its influence on the stability of the dam. For design purposes，it emerges that availability of reliable data on the hydraulic permeability of rock foundations and a computationally advanced distinct element modeling might lead to the acceptance of loads significantly higher than the more conservative estimations obtained from equilibrium analyses.Key words：hydraulic engineering；distinct element method(DEM)；hydraulic uplift；dam stability；gravity damCLC number：TV 642 Document code：A Article ID：1000–6915(2008)08–1554–15 坝底水浮力对重力坝稳定性的影响分析UTILI Stefano1，2，尹振宇2，蒋明镜3(1. 米兰理工大学土木工程系，意大利米兰 20133；2. 斯特莱斯克莱德大学土木工程系，英国格拉斯堡G4ONG；3. 同济大学地下建筑与工程系，上海 200092)摘要：着重研究一个典型的混凝土重力坝的坝底水浮力对大坝稳定性的影响，此大坝位于意大利的Cumbidanovu 岛。

燃料电池流场三角形和矩形比较

An analytical model for ﬂuid ﬂow and heat transferin a micro-heat pipe of polygonal shapeBalram Sumana,b,*,Prabhat KumarcaDepartment of Chemical Engineering and Materials Science,Mailbox #30,151Amundson Hall,421Washington Avenue,SE,University of Minnesota,Minneapolis 55455,MN,United StatesbSchool of Mathematics,University of Minnesota,Minneapolis 55455,MN,United States cDepartment of Food Science,North Carolina State University,Raleigh 27695,NC,United StatesReceived 18January 2005;received in revised form 29March 2005Available online 11July 2005AbstractAn analytical model for ﬂuid ﬂow and heat transfer in a micro-heat pipe of polygonal shape is presented by utilizing a macroscopic approach.The coupled nonlinear governing equations for ﬂuid ﬂow,heat and mass transfer have been modiﬁed and have been solved analytically.The analytical model enables us to study the performance and the limita-tions of such a device and provides the analytical expressions for critical heat input,dry-out length and available cap-illary head for the ﬂow of ﬂuid.A dimensionless parameter,which plays an important role in predicting the performance of a micro-heat pipe,is obtained from the analytical model.The results predicted by the model compared with the published results in literature and good agreement has been obtained.The general and analytical nature of the simple model will have its applicability in the design of micro-heat pipes.Ó2005Elsevier Ltd.All rights reserved.Keywords:Micro-heat pipe;Capillary forces;Critical heat input;Dry-out length;Analytical model1.IntroductionMicro-grooved heat pipes have become one of the most promising cooling devices because of their high eﬃciency,reliability and cost eﬀectiveness.The applica-tions of micro-heat pipes have expanded from the ther-mal control of integrated electronic circuits packaging,laser diodes,photovoltaic cells,infrared (IR)detectorsand space vehicles to the removal of heat from the lead-ing edges of stator vanes in turbines and nonsurgical treatment of cancerous tissue.The current concepts in integrated circuit (IC)packaging are motivated by the development of metal oxide semi-conductor (MOS)and very large scale integration (VLSI)technologies,which require higher levels of device integration.The continuous increase in the device integration density leads to a rapid rise in the power generation from such device resulting in increased thermal gradient and higher mean operating temperature in the devices leading to their improper functioning.Thus,it is necessary to de-velop new thermal control schemes capable of removing heat from the electronic chip.In cases where large amounts of heat need to be removed,two-phase heat0017-9310/$-see front matter Ó2005Elsevier Ltd.All rights reserved.doi:10.1016/j.ijheatmasstransfer.2005.05.001*Corresponding author.Address:Department of Chemical Engineering and Materials Science,Mailbox #30,151Amund-son Hall,421Washington Avenue,SE,University of Minne-sota,Minneapolis 55455,MN,United States.Tel.:+16126256083;fax:+16126267246.E-mail address:suman@ (B.Suman).International Journal of Heat and Mass Transfer 48(2005)4498–4509/locate/ijhmttransfer can prove to be a better technique.Therefore,a micro-grooved heat pipe is one of the promising options for micro-electronics cooling.Although the use of micro-heat pipes for enhanced heat transfer is becoming more common,the exact nature of liquid evaporation from the corners of a micro-heat pipe and the associated capillary pumping capacity has not been investigated in detail.The analytical model of micro-heat pipes is neces-sary for the complete understanding of the transport processes and the design of such devices.Cotter[1]ﬁrst proposed the concept of micro-heat pipe as a wickless heat pipe for uniform heat distribution in electronic chip.The micro-grooved heat pipe isﬁlled with a workingﬂuid and the pipe is sealed.Heatﬂux is applied to a portion of the heat pipe,called evapora-tive section,to vaporize the liquid in that region.TheNomenclaturea side length of a regular polygon or smallerside of a rectangle,ma1larger side length of a rectangle,mA cs area of cross-section forﬂuidﬂow,m2A l total liquid area,m2A0 l liquid area of one corner,m2B performance factorB1constant in expression for A lB0 1constant in expression for RÃin condensing regionB0 2constant in expression for RÃin adiabatic regionB0 3constant in expression for RÃin evaporative regionB2constant in expression for d RÃd XÃf friction factorf1nondimensional coordinate of the junction of condensing and adiabatic sectionsf2nondimensional coordinate of the junction of adiabatic and evaporative sectionsg acceleration due to gravity,m/s2K0constant in expression for B2L length of heat pipe,mL d dry-out length,mL h half of total wetted length,mK s thermal conductivity,W/(mK)m constant in the expression for generalized heatﬂuxn number of sides of a heat pipe polygonN Re Reynolds numberP l liquid pressure,N/m2PÃl nondimensional liquid pressureP R reference pressure,N/m2P vo pressure in vapor region,N/m2 Q net heat supplied,WQ in heat input,WQ0heat input,WQ0 c heatﬂux in the condensing region,W/m2Q cr critical heatﬂux,W/m2Q0 e heatﬂux in the evaporative region,W/m2Q v heatﬂux for vaporization of liquid,W/m2 R radius of curvature,mRÃnondimensional radius of curvature RÃ1nondimensional radius of curvature at f1RÃ2nondimensional radius of curvature at f2R0reference radius,mRe rectangleR l meniscus surface area per unit length,mT con temperature at the cold end,°CTr triangleT R reference temperature,°CT s temperature of substrate,°CTÃsdimensionless substrate temperatureV g axial vapor velocity,m/sV l axial liquid velocity,m/sVÃgnondimensional vapor velocityVÃlnondimensional liquid velocityV R reference liquid velocity,m/sW b perimeter of the heat pipe polygon,mx coordinate along the heat pipe,mXÃnondimensional coordinate along heat pipeXÃdnondimensional coordinate of the onset ofdry-regionGreek symbolsa half apex angle of polygon,radianb inclination of substrate with horizontal,radianc contact angle,radian/curvature,mÀ1k latent heat of vaporization of coolant liquid, J/kgl viscosity of coolant liquid,kgmÀ1secÀ1q l density of coolant liquid,kg/m3q g density of vapor,kg/m3r surface tension of coolant liquid,N/ms w wall shear stress,N/m2g,n,f variables used for integrationSubscriptsc condenser sectiond onset of dry-out regione evaporative sectiong vaporl coolant liquidB.Suman,P.Kumar/International Journal of Heat and Mass Transfer48(2005)4498–45094499vapor is pushed towards the condenser section.The cap-illary pressure and the temperature diﬀerence between the evaporative and condenser sections promote theﬂow of the workingﬂuid from the condenser back to the evaporator through the corner regions.Suman and Hoda[2]showed that the sharp angled corners are neces-sary for a good heat pipe operation.Swanson and Herdt [3]showed that the meniscus recession in the evaporative section causes a reduction in the meniscus radius causing an increase in the capillary pressure.Theﬂow ofﬂuid is primarily governed by the pressure diﬀerence at the liquid–vapor interface.The pressure diﬀerence at the inter-face is a function of the radius of curvature of the liquid meniscus,surface tension and wettability of the coolant liquid and substrate system[4–13].Micro-heat pipes,which could be embedded directly onto the silicon substrate of an integrated circuit,have been investigated in several studies conducted by Mallik and Peterson[14],Mallik et al.[15]and Peterson et al.[16].They have also reported experimental data taken on several water-charged micro-heat pipes with a cross-sectional dimension of about one millimeter.Ba-bin et al.[17]have developed a steady-state micro-heat pipe model to quantify heat transport capacity.Longtin [18]has developed a one-dimensional steady-state model of the evaporative and adiabatic sections of a micro-heat pipe to calculate workingﬂuid pressure,velocity and ﬁlm thickness along the length of the pipe.Peterson and his co-workers[19,20]have successfully used the Young–Laplace equation to describe the internalﬂuid dynamics of micro-heat pipes.The minimum meniscus radius and maximum heat transport in triangular grooves have also been studied[20].A detailed thermal analysis and various limitations of a micro-heat pipe have been done by Khrustalev and Faghri[21].Xu and Carey[22]have developed an analytical model for evaporation from a V-shaped micro-groove surface assuming that the evaporation takes place only from the thinﬁlm region of the meniscus.Ravikumar and DasGupta[23]have presented a model for evaporation from V-shaped ter,researchers have investigated the concept of using micro-heat pipe as an eﬀective heat spreader[24,25].Determination of dry-out length of a micro-heat pipe has also been investi-gated[26,27].Catton and Stores[28]presented one-dimensional semi-analytical model for the prediction of wetted length in inclined triangular capillary groove. They have introduced the concept of accommodation theory for change in the radius of curvature at the liquid–vapor interface.Suman et al.[29]have presented a generalized model of a micro-grooved heat pipe of polygonal shape.They have performed detailed study of triangular and rectangular heat pipes.Suman et al. have also presented semi-transient modeling of a micro-grooved heat pipe of polygonal shape[30]and transient analysis of a V-shaped micro-heat pipe[31].Technical issues related to micro-heat pipes investigated so far include liquid distribution and charge optimi-zation by Duncan and Peterson[33]and Mallik and Peterson[34,35],interfacial thermodynamics in micro-heat pipe capillary structures by Swanson and Peterson [36,37]and micro-heat pipe transient behavior by Wu et al.[38].Peterson[39]and Cao and Faghri[40]have done reviews of the literature on micro-heat pipes.In the present study,the coupled nonlinear governing equations forﬂuidﬂow,heat transfer and mass transfer developed for a general model of a micro-grooved heat pipe of any polygonal shape have been solved analyti-cally.The axialﬂow of the liquid as a result of the change in the radius of curvature has been modeled. The expressions for radius of curvature,liquid velocity, vapor velocity,liquid pressure,evaporative heatﬂux, critical heat input,dry-out length and available capillary head have been derived.The results predicted by the model have been successfully compared with the pub-lished results in literature.2.TheoryThe system in the present study consists of axially grooved-micro-heat pipe with grooves of any polygonal shape.Though the model equations in terms of the shape of the grooves are general in nature,an equilateral triangular and a rectangular heat pipe have been studied as test cases.The liquid is pushed from the cold end to the hot end due to decrease in the radius of curvature caused by the intrinsic meniscus receding into the cor-ner.The liquidﬂows along the corners and the vapor passes through the open space(Fig.1).The hot end and the cold end denote the farthest end of the evapora-tive and the condenser sections respectively.Decrease in the radius of curvature results decreases the liquid pres-sure,which is the driving force for theﬂow.The liquid ﬁlm gradually becomes thinner and more curved(lower radius of curvature)as the liquid recedes further towards the apex of the corner.It has been shown earlier[23]that the axialﬂow is primarily caused by capillary forces.The capillary forces are directly proportional to surface tension of the liquid and inverse of the radius of curvature.The radius of curvature of the liquidﬁlm is a function of the axial distance from the cold end.The model presented develops the governing equations forﬂuidﬂow and heat transfer and relates them to the capillary forces present in the system.The model equations are developed for all the three sections i.e.,evaporative,adiabatic and con-denser encompassing the complete heat pipe.The governing model equations are derived with the following assumptions:(i)one dimensional steady incompressibleﬂow along the length of heat pipe;(ii) uniform distribution of heat input;(iii)negligible heat4500 B.Suman,P.Kumar/International Journal of Heat and Mass Transfer48(2005)4498–4509dissipation due to viscosity;(iv)constant pressure in the vapor region in the operating range of temperature;(v)one dimensional temperature variation along the length of heat pipe;(vi)negligible shear stress at the liquid vapor interface.The radius of curvature (R )at the cold end is a con-stant,which can be calculated from the geometry of the groove since the cold end is completely ﬁlled i.e.R =R 0at the cold end.It is presented in the Appendix A .The evaluation of relevant geometrical parameters for diﬀer-ent polygonal shape is also presented in Appendix A .One corner of a section of heat pipe of any polygonal shape of length D x is shown in Fig.2.The radius of cur-vature,liquid velocity and pressure vary along the length of the heat pipe.The wall shear stress (s w )acts against the ﬂow of the liquid.The steady-state momentum balance in diﬀerential form is given as q l A l V ld ðV l Þd xþA ld P ld x þ2L h s w þq l g sin ðb ÞA l ¼0ð1ÞThe ﬁrst term in the above equation is the convectivemomentum change,the second term is the pressure force acting on the volume element,the third term is the wall shear force and the fourth term is gravitational force.Eq.(1)is valid for all three regions,namely evaporative,adiabatic and condensing regions.The contribution of the convective term has been found to be negligible using the order of magnitude analysis.It has also been shown [41]that the eﬀect of gravity is negligible and heat pipes are widely used at places where gravity is not important.Therefore,after neglecting convective and gravity terms Eq.(1)can be written as A ld P ld xþ2L h s w ¼0ð2ÞThe liquid pressure can be estimated from the Young–Laplace equation as P l ÀP vo ¼Àr ð3Þwhere P l is the liquid pressure,P vo is the pressure in the vapor region and R is the radius of curvature of liquid meniscus at any location,x .The liquid velocity can be given asV l ¼R x0W b Q 0cðf Þd f l l in condensing section V l ¼Ql l in adiabatic section V l ¼R L xW b Q 0e ðf Þd f q l k A lin evaporative section 9>>>>>>>>=>>>>>>>>;ð4ÞB.Suman,P.Kumar /International Journal of Heat and Mass Transfer 48(2005)4498–45094501where W b is perimeter of the polygon.Q 0c is the heat ﬂux in the condensing section,which is negative.Q is the to-tal heat input to the heat pipe.Q 0e is the heat ﬂux in the evaporative section,which is positive.Eq.(3)is derived by writing the mass balances in all the three sections.Similarly,the vapor velocity can be given as V g ¼R x 0W b Q 0cðf Þd f g cs l in condensing section V g ¼Qq g k ðA cs ÀA l Þin adiabatic section V g ¼R L x W b Q 0e ðf Þd f q g k ðA cs ÀA l Þin evaporative section 9>>>>>>>>=>>>>>>>>;ð5ÞThe supplied heat raises the temperature of the liquid packet and also evaporates the liquid.Ravikumar and DasGupta [23]showed that the sensible heat content is very small as compared to the latent heat for vaporiza-tion and condensation.Hence,the sensible heat content can be neglected.Thus,the energy balance in the volume element can be expressed asQ v ¼ÀQ 0c W bl in condensing section Q v ¼0in adiabatic section Q v ¼Q 0e W bR lin evaporative section 9>>>>=>>>>;ð6ÞThe steady-state energy balance in the substrate is given byA cs K s d 2T s 2ÀQ 0c ðx ÞW b ¼0in condensing section A cs K sd 2T sd x 2¼0in adiabatic section A cs K s d 2T s d x2ÀQ 0e ðx ÞW b ¼0in evaporative section 9>>>>>>=>>>>>>;ð7Þwhere the ﬁrst term is the net change in the conductive heat in the control volume and the second term is the heat taken up by the coolant liquid.2.1.Boundary conditionsThe radius of curvature of the liquid meniscus at the cold end (x =0),R 0is a constant since the cold end is completely ﬁlled.It can be obtained from the geometry for a known contact angle system.Its calculation has been presented in Appendix A .At x ¼0;R ¼R 0;T s ¼T conAt x ¼f 1L ;d T s d x x ¼f 1L¼0At x ¼f 2L ;d T s d x x ¼f 2L¼0Atx ¼L ;Q in ¼ÀK s A csd T s d x x ¼L2.2.NondimensionalizationEqs.(2)–(7)are nondimensionalized using the following expressions:friction factor,f =K 0/N Re hydraulic diameter,D h =4A l /(2L h ),Reynolds number,N Re =D h q l V l /l ,wall shear stress,s w =q V 2f /2,reference velocity,V R ¼Q =ðq R 20k Þ,reference pressure,P R =r /R 0,reference temperature,T R =T con .The dimensionless para-meters are deﬁned as follows:R *(dimensionless radius of curvature)=R /R 0,X *(dimensionless position)=x /L ,V Ãl (dimensionless liquid velocity)=V l /V R ,V Ãg (dimensionless vapor velocity)=V g /V R ,P Ãl(dimen-sionless liquid pressure)=P l /P R ,T Ãs (dimensionless sub-strate temperature)=T s /T R .K 0is used in the expression of friction factor (f )and is a constant for a speciﬁc geometry (13.33for triangle and 15.55for rectangle with side ratio 1:2,[32,42]).R 0is the radius of curvature at the cold end and is a function of side length,contact an-gle for the substrate–coolant system,and apex angle of the polygon.After nondimensionalization and rearrang-ing,Eqs.(2),(3)and (7)can be written as d R ÃÃ¼ÀB 2V R V Ãl L 0ð8Þwhere B 2¼l K 0cos 2ða þc Þ2sin 2ah cot ða þc Þcos ða þc Þsin csin aþf cot ða þc ÞÀu =2g i Its derivation has been presented in Appendix A .P Ãl ¼P voP RÀR Ãð9Þd 2T Ãsd X Ã2ÀQ 0c ðX ÃÞW b L 2T R A cs K s ¼0in condensing section d 2T sd X Ã2¼0in adiabatic section d 2T Ãs d XÃÀQ 0e ðX ÃÞW b L 2R cs s ¼0in evaporative section 9>>>>>>>>=>>>>>>>>;ð10ÞThe dimensionless boundary condition can be written as at X Ã¼0;R Ã¼1;T Ãs ¼1at X Ã¼f 1;d T Ãs d X ÃX Ã¼f 1¼0at X Ã¼f 2;d T Ãs Ã X Ã¼f 2¼0atX Ã¼1;Q in ¼ÀK s A cs T R L d T Ãs d X Ã X Ã¼1Using above boundary condition,the variations ofdimensionless radius of curvature can be obtained if the expression for V Ãl is known.The temperature proﬁle4502 B.Suman,P.Kumar /International Journal of Heat and Mass Transfer 48(2005)4498–4509of the substrate can be obtained analytically using Eq.(10)and the above boundary conditions since the evap-orative and condensing heatﬂuxes are function of posi-tion.The expressions for liquid velocity,vapor velocity, dimensionless radius of curvature,liquid pressure and evaporative heatﬂux for all three sections have been pre-sented in next sections.2.2.1.The condensing sectionVÃl ¼R XÃW b Q0cðgÞd gV R q l k A lð11ÞVÃg ¼R XÃW b Q0cðgÞd gR g cs lð12ÞRÃ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1À3B01Z XÃZ gQ0cðnÞd n3sd gð13ÞwhereB0 1¼B2L2W br R3qlk B1PÃl ¼P voP RÀﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1À3B01Z XÃZ gQ0cðnÞd n3sd gð14ÞQv ¼ÀQ0cW bR lð15Þ2.2.2.The adiabatic sectionVÃl ¼QV R q l k A lð16ÞVÃg ¼QV R q g kðA csÀA lÞð17ÞRÃ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃðRÃ1Þ3À3QB02ðXÃÀf1Þ3qð18ÞwhereRÃ1¼RÃðf1Þ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1À3B01Z f1Z gQ0cðnÞd n3sd gandB0 2¼B2Lr R3qlk B1PÃl ¼P voRÀﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃðRÃ1Þ3À3QB02ðXÃÀf1Þ3qð19ÞQv¼0ð20Þ2.2.3.The evaporative sectionVÃl ¼R XÃW b Q0eðgÞd gV R q l k A lð21ÞVÃg ¼R XÃW b Q0cðgÞd gV R qgkðA csÀA lÞð22ÞRÃ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃðRÃ2Þ3À3B03Z XÃf2Z1gQ0eðnÞd n3sd gð23ÞwhereRÃ2¼RÃðf2Þ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃðRÃ1Þ3À3B02ðXÃÀf1Þ3qandB03¼B2L2W br R3q l k B1¼B01PÃl¼P voRÀﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃðRÃ2Þ3À3B03Z XÃZ gQ0eðnÞd n3sd gð24ÞQv¼Q0eW bR lð25Þ2.3.Critical heat input and dry-out lengthThe value of R*at the hot end can predict the max-imum heat input for the system known as the operatinglimit or the critical heat input of the heat pipe.The crit-ical heat input is deﬁned as the heat input for which theﬂow due to the curvature change is not able to meet therequirement ofﬂuid evaporation at the hot end i.e.suf-ﬁcient coolantﬂuid is not available for evaporation atthe hot end.For critical heat input,as noﬂuid is avail-able for evaporation,the radius of curvature of theliquid meniscus at the hot end reaches a value very closeto zero.Then,the device approaches its operating limit.Therefore,any heat input higher than the critical heatinput propagates the dry region starting from the hotend towards the cold end.Based on this principle,theexpressions for the critical heat input and the dry-outlength have been developed.The critical heat input is the heat input for which theradius of curvature(R*)of the liquid meniscus at the hotend(X*=1)reaches a value very close to zero and thedevice approaches its operating limit.Mathematically,this can be represented as follows:For Q=Q cr,R*=0at X*=1Hence,by solving Eqs.(13),(18)and(23)for knownheatﬂux distributions in evaporative and condensingsections,the expression for the critical heat input canbe obtained.By taking constant heatﬂuxes in evapora-tive and condensing regions(Q0c¼Q1band Q0e¼Qð1Àf2ÞW b L),the expression for the critical heat input canbe given as follows:Qcr¼2B1rq l k R3221ð26ÞBy taking Q0c¼Qðmþ1ÞW bðf1LÞmþ1ðf1ÀXÃÞm and Q0e¼Qðmþ1ÞW bðLð1Àf2ÞÞmþ1ÂðXÃÀf2Þm,the expression for critical heat input is obtainedasB.Suman,P.Kumar/International Journal of Heat and Mass Transfer48(2005)4498–45094503Q cr ¼rq l k B 1ðR 0Þ33B 2L ðf 2Àf 1Þþ221m ð27ÞThe analytical model presented here can be used to cal-culate the dry-out length (L d )for a set of process vari-ables.The radius of curvature at the hot end for the critical heat input becomes very close to zero.The cap-illary pumping becomes less than the rate of evaporation for heat input higher than the critical heat.This propa-gates the dry region starting from the hot end towards the cold end.The dry region of the heat pipe is known as dry-out length.If Q <Q cr ,the heat pipe is working without generating any dry-region.If the heat input is greater than the critical heat input and less than the heat required (Q 2)to onset the dry-out at the junction of the evaporative and the adiabatic sections i.e.R Ã2¼0,the dry-out region is only in the evaporative section.The general expression for Q 2can be obtained by equat-ing R Ã2to zero as follows:3B 02ðf 2Àf 1Þ¼1À3B 01Z f 10Z gQ 0ðn Þd n ð28ÞThe dry-out length can be calculated by evaluating X Ãd asfollows:3B 03Z X Ãdf 2Z 1gQ 0ðn Þd n d g ¼R 32ð29ÞandL d ¼L ð1ÀX Ãd Þð30ÞFor the constant heat input distributions in evapora-tive and condensing sections,the expressions for Q 2and the dry-out length can be given as follows:Q 2¼r B 1q l k R 303B 2L ðf 2Àf 1Þþðf 1Þ22ð1Àf 2Þ()ð31ÞL d ¼L ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1À2ðR Ã2Þ3ð1Àf 2Þr B 1R 30q l k 3B 2LQ þf 2Àðf 2Þ22()v u u t ð32ÞIf Q 2<Q <Q 1(input heat required to onset of dry-out at the junction of the adiabatic and condensing sec-tions),the dry-region is in the evaporative and the adia-batic sections.The expression for Q 1can be obtained by equating R Ã1to zero as1¼3B 01Zf 1ZgQ 0ðn Þd nd gð33ÞThe expression for the dry-out length can be obtainedusing Eq.(30)where X Ãd is given as follows:X Ãd ¼R 313B 02þf 1ð34ÞFor constant heat input distributions in evaporative and condensing sections,the expressions for Q 1and the dry-out length are given as follows:Q 1¼2B 1r R 30q l k 3B 2Lf 1ð35ÞL d ¼L 1ÀðR Ã1Þ302þf 1!ð36ÞIf Q >Q 1,the dry-out is in all three sections and it is calculated using Eq.(30)where X Ãd is given as follows:3B 01Z X ÃcZ g 0Q 0ðn Þd n d g ¼1ð37ÞFor constant heat input distribution,the expression fordry-out is given as follows:L d ¼L 1Àﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2B 1r R 30q l k 3B 2Lf 1s 0@1A ð38Þ2.4.Available capillary headCapillary pumping is the driving force for ﬂuid ﬂow in a micro-heat pipe.Therefore,it is useful to know the available capillary head for ﬂuid ﬂing the ana-lytical model,the expression for the available capillary head for ﬂuid ﬂow is given asFor the constant heat input in evaporative and con-densing sections,the expression for the available capil-lary head becomesH c ¼rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1À3B 2LQ ðf 1Àf 2þ1Þ2rq l k B 1ðR 0Þ33s ð40Þ3.Results and discussionTwo test cases are analyzed and the model is vali-dated with the numerical results of Suman et al.[29]H c ¼r1R Ãð1ÞÀ1R Ãð0Þ¼rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1À3B 01R f 10n R g 0Q 0ðn Þd n o d g À3B 02ðf 2Àf 1ÞÀ3B 03R 1f 2R 1g Q 0ðn Þd n n o 3r d g ð39Þ4504 B.Suman,P.Kumar /International Journal of Heat and Mass Transfer 48(2005)4498–4509。

人物外貌性格描写英语词汇

⼈物外貌性格描写英语词汇Expressions for Appearance（⼈物外貌描写）short ，tall ，overweight ，plump 丰满thin ，skinny太瘦了slim 苗条medium height 不胖不瘦tubby 矮胖muscular 强壮elegantly 优雅good-looking 长的好看plain 长的⼀般neatly dressed ⾐着⼲净整洁scruffily dressed ⾐着不整洁 smartly dressed: 穿着得体 well dressed :穿的漂亮figure⾝材体形:overweight 超重thin, slim 瘦fat, stout 肥胖的slender 苗条的tallish 有些⾼的; middle-sized ⾝⾼中等的; medium height 中等⾝材chubby (丰满)，plump (丰满), fat, long and slim, tall, massive, slender, average height，obese (过胖)，thick-set (矮⽽胖)，dwarfish (矮⼩)，short ，overweight ，thin ，skinny，medium height ，tubby （矮胖）， muscular,Hair发型:dark hair blonde ⾦发 curly, wavy hair 卷发 pony tail 马尾straight 直的jet-black 乌⿊的; fair ⾦⾊的; blonde 淡黄⾊的; golden ⾦黄⾊的; dyed 染⾊的curly卷发的; pigtails辫⼦; crew cut平头bald, baldhead秃头sparse, thick, straight, pitch-black, oily, dry, soft, wavy, wit h natural curls, light-brown, stand on end, with some hair tur ning gray, parted in the middle, combed to one side, combedinto a knot/a brush at the back, with hair let down to the eyebrows/shoulders, butch平头的，bobbed hair短发, with luster, braid, plait辫⼦hoary灰⽩的face:脸部:dimple 酒窝 oval face 鸭蛋脸 bald 秃顶的bobbed短卷发的 Wrinkled 起皱纹的Big-featured 五官较⼤的 Double-chinned 双下巴的 Slender 苗条的Sharp-featured 五官分明的 Well-featured 五官端正的 Rosy/ruddy红润的Freckled 有雀斑的 Well-Shaped 好看的 round 圆的; thin ; long 长的; square ⽅的。

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µ
h ¯ ω0 2
2 x2 i + πQj )
ij
(1)
Qµ (i) Q† µ (j )
(2) (3)
µ Qµ (i) = b2 x2 i Y2 (xi )
with dimensionless coordinate and momenta xi = ri /b, π i = pi b/h ¯ , and the harmonic oscillator (H.O.) length scale b = h ¯ /mω0 . 2
To evaluate Eq. (7) we have to keep in mind that according to the selection rules for H.O. wave functions only matrix elements with Nc = N − 2 will contribute to the sum over Nc . This is clear since the Qµ operator only connects the states with N to those with N ′ = N, N ± 2, and of these possible N ′ values only N ′ = Nc = N − 2 belongs to the set of core levels. Carrying out the sum over mc and µ in Eq. (7) we get
Typeset using REVTEX 1
I. INTRODUCTION
In a recent publication [1], Fayache, Sharon and Zamick compared the collective model result for a rotational band to the 0d-1s splitting obtained with the OXBASH program [2]. For the shell model calculations done with OXBASH they use a Hamiltonian consisting of a spherical harmonic oscillator and a residual interaction which is the isoscalar quadrupolequadrupole (Q-Q) in coordinate space only, i.e., they omit the Q · Q term in momentum space and allow for interactions between shells with ∆N = 2. From these shell model calculations the authors of Ref. [1] found that one third of the 0d-1s single particle (s.p.) splitting (6χ) comes from the interaction of the valence particle with the core while two thirds (12χ) come from the diagonal Q · Q interaction. The 0d-1s s.p. energy-splitting (18χ) is the same as the energy splitting between L=2 and L=0 rotational bands obtained with Elliott’s SU(3) model in the s-d shell [3]. A similar numerical relation was found in the f-p shell [1]. This is an interesting observation that deserves further study. In this paper we give an analytical proof of those numerical relations and show that they are particular cases of a general property of the Hamiltonian used. To this end we ﬁrst derive complete analytical expressions for the s.p. energies (these will be deﬁned more precisely in Sec. 2), and then derive the s.p. energy splittings in a major shell using the same Hamiltonian as in Ref. [1]. We consider the problem of how to obtain the s.p. energy splittings in a major shell with such an interaction which will preserve the SU(3) results More speciﬁcally, we use the Hamiltonian H= where Q (i) · Q(j ) =
Q1 where EN ℓ comes from the diagonal part of the Q-Q interaction (Q (i) · Q (i)) Q1 EN ℓ = −
χ < Nℓ|Q · Q|Nℓ > 2
(5)
or
Q1 4 EN ℓ = −4χ < Nℓ|x |Nℓ >, χ = χ
Q2 EN ℓ = 8χ ℓc 2
< N ℓ |x2 |N − 2 ℓc >
(2ℓc + 1)

0 0 0
ℓ 2 ℓc

2
(9)
For H.O. wave functions the matrix element in Eq. (9) is given by < N ℓ |x2 |N − 2 ℓ′ >= δℓ′ ,ℓ n n + ℓ + +δℓ′ ,ℓ−2 1 + δℓ′ ,ℓ+2 n (n − 1) 2 1 2 n+ℓ− 1 2 (10)
2
Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08855, USA (February 9, 2008)
Abstract
We present an analytical proof and a generalization of the Fayache-SharonZamick relation between single particle energy splittings and the SU(3) limit in Elliott’s model. 21.60.-n,21.10.-k
II. DEFINITION OF SINGLE PARTICLE ENERGIES AND ENERGY SPLITTINGS
To parallel the shell model calculations of Ref. [1], we consider the eﬀect of the residual Q-Q interaction when we have a closed core and put a valence nucleon in diﬀerent orbital (Nℓm) states of a major shell (N ) out of the closed core. In other words, we look for the s.p. energy splitting between Nℓ and Nℓ′ levels caused by the Q-Q interaction. To this end we write the s.p. energy of the Nℓ level as EN ℓ = h ¯ ω0 N + 3 Q1 Q2 + EN ℓ + EN ℓ 2 (4)
Q1 we obtain the following result for EN ℓ
3
Q1 EN ℓ = −4χ n(n − 1) + 4n n + ℓ +
3 2
+ n+ℓ+
3 2
n+ℓ+
5 2 (8)
= χ 2ℓ (ℓ + 1) − 3 2N 2 + 6N + 5 with n the principal quantum number (n = (N − ℓ)/2)(see for instance [4]).
Analytic expressions for the single particle energies with a quadrupole-quadrupole interaction and the relation to Elliott’s SU(3) model
E. Moya de Guerra1 , P. Sarriguren1 and L. Zamick2
1
Instituto de Estructura de la Materia, Consejo Superior de
arXiv:nucl-th/9703052v2 17 Jun 1997
Investigaciones Cient´ ıﬁcas, Serrano 123, E-28006 Madrid, Spain