Lie point symmetries and the geodesic approximation for the Schrodinger-Newton equations

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证明开普勒猜想的原文(英文)并附译后汉语

Thus it can seen, the general volume of two such rectangular parallelepipeds is just equal to the general volume of aforementioned the annular solid, for either general volume is equal to (√2-1)L3.
Thus it can computed, the ratio of the volume of the internal tangent sphere of cube N to the volume of cube K is equal to √2π/6, i.e. π /√18. This ratio is just that ratio which the Kepler’s conjecture mentions.
Abstract
Heap together equivalent spheres into a cube up to most possible, then variant general volumes of equivalent spheres inside the cube depend on variant arrangements of equivalent spheres fundamentally. This π/√18 which the Kepler’s conjecture mentions is the ratio of the general volume of equivalent spheres under the maximum to the volume of the cube. We will do a closer arrangement of equivalent spheres inside a cube. Further let a general volume of equivalent spheres to getting greater and greater, up to tend upwards the super-limit, in pace with which each of equivalent spheres is getting smaller and smaller, and their amount is getting more and more. We will prove the Kepler’s conjecture by such a way in this article.

gravity

energy need be considered here.The ﬁrst is the potential energy,which an object has by virtue of its position rela-tive to the origin of a force.The second is the work done against the action of the force during a change in position.For example,when Newton’s apple is on the tree it has a higher potential energy than when it lies on the ground. It falls because of the downward force of gravity and loses potential energy in doing so.To compute the change in potential energy we need to raise the apple to its original position.This requires that we apply a force equal and opposite to the gravitational attraction on the apple and, because this force must be moved through the distance the apple fell,we have to expend energy in the form of work.If the original height of the apple above ground level was h and the value of the force exerted by gravity on the apple is F,the force we must apply to put it back is (ϪF). Assuming that F is constant through the short distance of its fall,the work expended is (ϪF)h.This is the increase in potential energy of the apple,when it is on the tree.More generally,if the constant force F moves through a small distance d r in the same direction as the force,the work done is d WϭF d r and the change in potential energy d Epis given by(2.3)In the more general case we have to consider motions and forces that have components along three orthogonal axes.The displacement d r and the force F no longer need to be parallel to each other.We have to treat F and d r as vectors.In Cartesian coordinates the displacement vector d r has components (d x,d y,d z) and the force has compo-nents (Fx ,Fy,Fz) along each of the respective axes.Thed Ep ϭϪd WϭϪF d rFig. 2.4Geometries for the gravitational attraction on (a) two point masses, (b) a point mass outside a sphere, and (c) a point mass on the surface of a sphere.of the gravitational potential of a continuous mass distribution.straight line unless compelled to change that state by forces acting on it.The continuation of a state of motion is by virtue of the inertia of the body.A framework in which this law is valid is called an inertial system .For example,when we are travelling in a car at constant speed,we feel no disturbing forces;reference axes ﬁxed to the moving vehicle form an inertial frame.If tra ﬃc condi-tions compel the driver to apply the brakes,we experience decelerating forces;if the car goes around a corner,even at constant speed,we sense sideways forces toward the outside of the corner.In these situations the moving car is being forced to change its state of uniform rectilinear motion and reference axes ﬁxed to the car form a non-inertial system .Motion in a circle implies that a force is active that continually changes the state of rectilinear motion.Newton recognized that the force was directed inwards,towards the center of the circle,and named it the cen-tripetal (meaning “center-seeking”) force.He cited the example of a stone being whirled about in a sling.The inward centripetal force exerted on the stone by the sling holds it in a circular path.If the sling is released,the restraint of the centripetal force is removed and the inertia of the stone causes it to continue its motion at the point of release.No longer under the inﬂuence of the restraining force,the stone ﬂies o ﬀin a straight line.Arguing that the curved path of a projectile near the surface of the Earth was due to the e ﬀect of gravity,which caused it constantly to fall toward the Earth,Newton postulated that,if the speed of the projectile were exactly right,it might never quite reach the Earth’s surface.If the projectile fell toward the center of the Earth at the same rate as the curved surface of the Earth fell away from it,the projectile would go into orbit around the Earth.Newton suggested that the Moon was held inFig. 2.6(a) Equipotential surfaces of a spherical mass form a set of concentric spheres. (b) The normal to the equipotential surface deﬁnes the vertical direction; the tangential plane deﬁnes the horizontal.orbit around the Earth by just such a centripetal force, which originated in the gravitational attraction of the Earth.Likewise,he visualized that a centripetal force due to gravitational attraction restrained the planets in their circular orbits about the Sun.The passenger in a car going round a corner experi-ences a tendency to be ﬂung outwards.He is restrained in position by the frame of the vehicle,which supplies the necessary centripetal acceleration to enable the pas-senger to go round the curve in the car.The inertia of the passenger’s body causes it to continue in a straight line and pushes him outwards against the side of the vehicle. This outward force is called the centrifugal force.It arises because the car does not represent an inertial reference frame.An observer outside the car in a ﬁxed (inertial) coordinate system would note that the car and passenger are constantly changing direction as they round the corner.The centrifugal force feels real enough to the passenger in the car,but it is called a pseudo-force,or inertial force.In contrast to the centripetal force,which arises from the gravitational attraction,the centrifugal force does not have a physical origin,but exists only because it is being observed in a non-inertial reference frame.2.3.2.1Centripetal accelerationThe mathematical form of the centripetal acceleration for circular motion with constant angular velocity v about a point can be derived as follows.Deﬁne orthogo-nal Cartesian axes x and y relative to the center of the circle as in Fig.2.7a.The linear velocity ␯at any point where the radius vector makes an angle uϭ(v t) with the x-axis has componentsvxϭϪv sin(v t) ϭϪr v sin(v t)v y ϭv cos(v t) ϭr v cos(v t)(2.17) The x- and y-components of the acceleration areobtained by diﬀerentiating the velocity components with respect to time.This givesaxϭϪv v cos(v t) ϭϪr v2cos(v t)a y ϭϪv v sin(v t) ϭϪr v2sin(v t)(2.18) These are the components of the centripetal accelera-tion,which is directed radially inwards and has the mag-nitude v2r(Fig.2.7b).2.3.2.2Centrifugal acceleration and potentialIn handling the variation of gravity on the Earth’s surface we must operate in a non-inertial reference frame attached to the rotating Earth.Viewed from a ﬁxed,external inertial frame,a stationary mass moves in a circle about the Earth’s rotation axis with the same rotational speed as the Earth.Fig. 2.7(a) Components vx and vyof the linear velocity v where theradius makes an angle u(v t) with the x-axis, and (b) the components ax and ayof the centripetal acceleration, which is directed radially inward.Fig. 2.8The outwardly directed centrifugal acceleration ac at latitude lon a sphere rotating at angular velocity v.The mass of the Moon is 0.0123 that of the Earth and the distance between the centers is 384,100km.These ﬁgures give dϭ4600km,i.e.,the center of revolution of the Earth–Moon pair lies within the Earth.It follows that the paths of the Earth and the Moon around the Sun are more complicated than at ﬁrst appears.The elliptical orbit is traced out by the barycen-ter of the pair (Fig.2.9).The Earth and Moon follow wobbly paths,which,while always concave towards the Sun,bring each body at diﬀerent times of the month alternately inside and outside the elliptical orbit.To understand the common revolution of the Earth–Moon pair we have to exclude the rotation of the Earth about its axis.The “revolution without rotation”is illustrated in Fig.2.10.The Earth–Moon pair revolves about S,the center of mass.Let the starting positions be as shown in Fig.2.10a.Approximately one week later the Moon has advanced in its path by one-quarter of a revolu-tion and the center of the Earth has moved so as to keep the center of mass ﬁxed (Fig.2.10b).The relationship isFig. 2.9Paths of the Earth and Moon, and their barycenter, around the Sun.Fig. 2.10Illustration of the “revolution without rotation” of the Earth–Moon pair about their common center of mass at S.At points D and D Јthe direction of the gravitationalacceleration due to the Moon is not exactly parallel to the line of centers of the Earth–Moon pair.The residual tidal acceleration is almost along the direction toward the center of the Earth.Its e ﬀect is to lower the free surface in this direction.The free hydrostatic surface of the Earth is an equipo-tential surface (Section 2.2.3),which in the absence of the Earth’s rotation and tidal e ﬀects would be a sphere.The lunar tidal accelerations perturb the equipotential surface,raising it at A and B while lowering it at D and D Ј,as in Fig.2.11a.The tidal deformation of the Earth produced by the Moon thus has an almost prolate ellipsoidal shape,like a rugby football,along the Earth–Moon line of centers.The daily tides are caused by superposing the Earth’s rotation on this deformation.In the course of one day a point rotates past the points A,D,B and D Јand an observer experiences two full tidal cycles,called the semi-diurnal tides.The extreme tides are not equal at every latitude,because of the varying angle between the Earth’s rotational axis and the Moon’s orbit (Fig.2.11b).At the equator E the semi-diurnal tides are equal;at an intermediate latitude F one tide is higher than the other;and at latitude G and higher there is only one (diurnal) tide per day.The di ﬀerence in height between two successive high or low tides is called the diurnal inequality .In the same way that the Moon deforms the Earth,so the Earth causes a tidal deformation of the Moon.In fact,the tidal relationship between any planet and one of its moons,or between the Sun and a planet or comet,can be treated analogously to the Earth–Moon pair.A tidal acceleration similar to Eq.(2.31) deforms the smaller body;its self-gravitation acts to counteract the deforma-tion.However,if a moon or comet comes too close to the planet,the tidal forces deforming it may overwhelm the gravitational forces holding it together,so that the moon or comet is torn apart.The separation at which this occurs is called the Roche limit (Box 2.1).The material of a disintegrated moon or comet enters orbit around the planet,forming a system of concentric rings,as around the great planets (Section 1.1.3.3).2.3.3.2Tidal effect of the SunThe Sun also has an inﬂuence on the tides.The theory of the solar tides can be followed in identical manner to the lunar tides by again applying the principle of “revolution without rotation.”The Sun’s mass is 333,000 times greater than that of the Earth,so the common center of mass is close to the center of the Sun at a radial distance of about 450km from its center.The period of the revolution is one year.As for the lunar tide,the imbalance between gravita-tional acceleration of the Sun and centrifugal acceleration due to the common revolution leads to a prolate ellip-soidal tidal deformation.The solar e ﬀect is smaller than that of the Moon.Although the mass of the Sun is vastlyFig. 2.11(a) The relationships of the centrifugal, gravitational and residual tidal accelerations at selected points in the Earth. (b) Latitude effect that causes diurnal inequality of the tidal height.greater than that of the Moon,its distance from the Earth is also much greater and,because gravitational accelera-tion varies inversely with the square of distance,the maximum tidal eﬀect of the Sun is only about 45% that of the Moon.2.3.3.3Spring and neap tidesThe superposition of the lunar and solar tides causes a modulation of the tidal amplitude.The ecliptic plane is deﬁned by the Earth’s orbit around the Sun.The Moon’sorbit around the Earth is not exactly in the ecliptic but isinclined at a very small angle of about 5Њto it.For discus-sion of the combination of lunar and solar tides we can assume the orbits to be coplanar.The Moon and Sun each produce a prolate tidal deformation of the Earth,but the relative orientations of these ellipsoids vary during one month (Fig.2.12).At conjunction the (new)Moon is on the same side of the Earth as the Sun,and the ellipsoidal deformations augment each other.The same is the case half a month later at opposition,when the (full)Moon is on the opposite side of the Earth from the Sun.The unusually high tides at opposition and conjunction are called spring tides .In contrast,at the times of quadra-ture the waxing or waning half Moon causes a prolate ellipsoidal deformation out of phase with the solar defor-mation.The maximum lunar tide coincides with the minimum solar tide,and the e ﬀects partially cancel each other.The unusually low tides at quadrature are called neap tides .The superposition of the lunar and solar tides causes modulation of the tidal amplitude during a month (Fig.2.13).2.3.3.4Effect of the tides on gravity measurementsThe tides have an e ﬀect on gravity measurements made on the Earth.The combined e ﬀects of Sun and Moon cause an acceleration at the Earth’s surface of approxi-mately 0.3mgal,of which about two-thirds are due to the Moon and one-third to the Sun.The sensitive modern instruments used for gravity exploration can readily detect gravity di ﬀerences of 0.01mgal.It is necessary to compen-sate gravity measurements for the tidal e ﬀects,which vary with location,date and time of day.Fortunately,tidal theory is so well established that the gravity e ﬀect can be calculated and tabulated for any place and time before beginning a survey.2.3.3.5Bodily Earth-tidesA simple way to measure the height of the marine tide might be to ﬁx a stake to the sea-bottom at a suitably shel-tered location and to record continuously the measured water level (assuming that confusion introduced by wave motion can be eliminated or taken into account).The observed amplitude of the marine tide,deﬁned by the dis-placement of the free water surface,is found to be about 70% of the theoretical value.The di ﬀerence is explained by the elasticity of the Earth.The tidal deformation cor-responds to a redistribution of mass,which modiﬁes the gravitational potential of the Earth and augments the ele-vation of the free surface.This is partially counteracted by a bodily tide in the solid Earth,which deforms elasti-cally in response to the attraction of the Sun and Moon.The free water surface is raised by the tidal attraction,but the sea-bottom in which the measuring rod is implanted is also raised.The measured tide is the di ﬀerence between the marine tide and the bodily Earth-tide.In practice,the displacement of the equipotential surface is measured with a horizontal pendulum,which reacts to the tilt of the surface.The bodily Earth-tides also a ﬀect gravity measurements and can be observed with sensitive gravimeters.The e ﬀects of the bodily Earth-tides are incorporated into the predicted tidal cor-rections to gravity measurements.2.3.4Changes in Earth’s rotationThe Earth’s rotational vector is a ﬀected by the gravita-tional attractions of the Sun,Moon and the planets.The rate of rotation and the orientation of the rotational axis change with time.The orbital motion around the Sun is also a ﬀected.The orbit rotates about the pole to the plane of the ecliptic and its ellipticity changes over long periods of time.2.3.4.1Effect of lunar tidal friction on the length of the dayIf the Earth reacted perfectly elastically to the lunar tidal forces,the prolate tidal bulge would be aligned along the line of centers of the Earth–Moon pair (Fig.2.14a).However,the motion of the seas is not instantaneous and the tidal response of the solid part of the Earth is partly anelastic.These features cause a slight delay in the time when high tide is reached,amounting to about 12minutes.In this short interval the Earth’s rotation carries the line of the maximum tides past the line of centers by a small angle of approximately 2.9Њ(Fig.2.14b).A point on the rotating Earth passes under the line ofmaximumFig. 2.12The orientations of the solar and lunar tidal deformations of the Earth at different lunar phases.tides 12 minutes after it passes under the Moon.The small phase di ﬀerence is called the tidal lag.Because of the tidal lag the gravitational attraction of the Moon on the tidal bulges on the far side and near side of the Earth (F 1and F 2,respectively) are not collinear (Fig.2.14b).F 2is stronger than F 1so a torque is produced in the opposite sense to the Earth’s rotation (Fig.2.14c).The tidal torque acts as a brake on the Earth’s rate of rotation,which is gradually slowing down.The tidal deceleration of the Earth is manifested in a gradual increase in the length of the day.The e ﬀect is very small.Tidal theory predicts an increase in the length of the day of only 2.4 milliseconds per century.Observations of the phenomenon are based on ancient historical records of lunar and solar eclipses and on telescopically observed occultations of stars by the Moon.The current rate of rotation of the Earth can be measured with very accurate atomic clocks.Telescopic observations of the daily times of passage of stars past the local zenith are recorded with a camera controlled by an atomic clock.These observations give precise measures of the mean value and ﬂuctuations of the length of the day.The occurrence of a lunar or solar eclipse was a momentous event for ancient peoples,and was duly recorded in scientiﬁc and non-scientiﬁc chronicles.Untimed observations are found in non-astronomical works.They record,with variable reliability,the degree of totality and the time and place of observation.The unaided human eye is able to decide quite precisely just when an eclipse becomes total.Timed observations of both lunar and solar eclipses made by Arab astronomers around 800–1000 AD and Babylonian astronomers a thousand years earlier give two important groups of data (Fig.2.15).By comparing the observed times ofalignmentspring tideneap tidespring tide neap tideFig. 2.13Schematic representation of the modulation of the tidal amplitude as a result ofsuperposition of the lunar and solar tides.Fig. 2.14(a) Alignment of the prolate tidal bulge of a perfectly elastic Earth along the line of centers of the Earth–Moon pair. (b) Tidal phase lag of 2.9Њrelative to the line of centers due to the Earth’s partially anelastic response. (c) Tidal decelerating torque due to unequal gravitational attractions of the Moon on the far- and near-sided tidal bulges.of Sun,Moon and Earth with times predicted from the theory of celestial mechanics,the diﬀerences due to change in length of the day may be computed.A straight line with slope equal to the rate of increase of the length of the day inferred from tidal theory,2.4ms per century,con-nects the Babylonian and Arab data sets.Since the medieval observations of Arab astronomers the length of the day has increased on average by about 1.4ms per century.The data set based on telescopic observations covers the time from 1620 to 1980 AD.It gives a more detailed picture and shows that the length of the day ﬂuc-tuates about the long-term trend of1.4ms per century.A possible interpretation of the diﬀerence between the two slopes is that non-tidal causes have opposed the decelera-tion of the Earth’s rotation since about 950 AD.It would be wrong to infer that some sudden event at that epoch caused an abrupt change,because the data are equally compatible with a smoothly changing polynomial.The observations conﬁrm the importance of tidal braking,but they also indicate that tidal friction is not the only mecha-nism aﬀecting the Earth’s rotation.The short-term ﬂuctuations in rotation rate are due to exchanges of angular momentum with the Earth’s atmos-phere and core.The atmosphere is tightly coupled to the solid Earth.An increase in average global wind speed corresponds to an increase in the angular momentum of the atmosphere and corresponding decrease in angular momentum of the solid Earth.Accurate observations by very long baseline interferometry (see Section 2.4.6.6) conﬁrm that rapid ﬂuctuations in the length of the day are directly related to changes in the angular momentumrecordFig. 2.15Long-term changes in the length of the day deduced from observations of solar and lunar eclipses between 700 BC and 1980 AD (after Stephenson and Morrison, 1984).Earth’s rotation causes an increase in the Earth–Moon dis-tance,r L .At present this distance is increasing at about3.7cm yr Ϫ1.As a further consequence Eq.(2.37) shows that the Moon’s rate of revolution about the Earth (⍀L ) –and consequently also its synchronous rotation about its own axis – must decrease when r L increases.Thus,tidal friction slows down the rates of Earth rotation,lunar rota-tion,and lunar orbital revolution and increases the Earth–Moon distance.Eventually a situation will evolve in which the Earth’s rotation has slowed until it is synchronous with the Moon’s own rotation and its orbital revolution about the Earth.All three rotations will then be synchronous and equivalent to about 48 present Earth days.This will happen when the Moon’s distance from Earth is about 88times the Earth’s radius (r L ϭ88R ;it is presently equal to about 60R ).The Moon will then be stationary over the Earth,and Earth and Moon will constantly present the same face to each other.This conﬁguration already exists between the planet Pluto and its satellite Charon.2.3.4.3The Chandler wobbleThe Earth’s rotation gives it the shape of a spheroid,or ellipsoid of revolution.This ﬁgure is symmetric with respect to the mean axis of rotation,about which the moment of inertia is greatest;this is also called the axis of ﬁgure (see Section 2.4).However,at any moment the instantaneous rotational axis is displaced by a few meters from the axis of ﬁgure.The orientation of the total angular momentum vector remains nearly constant butthe axis of ﬁgure changes location with time and appears to meander around the rotation axis (Fig.2.16).The theory of this motion was described by Leonhard Euler (1707–1783),a Swiss mathematician.He showed that the displaced rotational axis of a rigid spheroid would execute a circular motion about its mean position,now called the Euler nutation .Because it occurs in the absence of an external driving torque,it is also called the free nuta-tion .It is due to di ﬀerences in the way mass is distributed about the axis of rotational symmetry and an axis at right angles to it in the equatorial plane.The mass distributions are represented by the moments of inertia about these axes.If C and A are the moments of inertia about the rota-tional axis and an axis in the equatorial plane,respectively,Euler’s theory shows that the period of free nutation is A /(C ϪA ) days,or approximately 305 days.Astronomers were unsuccessful in detecting a polar motion with this period.In 1891 an American geodesist and astronomer,S.C.Chandler,reported that the polar motion of the Earth’s axis contained two important com-ponents.An annual component with amplitude about 0.10 seconds of arc is due to the transfer of mass between atmosphere and hydrosphere accompanying the chang-ing of the seasons.A slightly larger component with amplitude 0.15 seconds of arc has a period of 435 days.This polar motion is now called the Chandler wobble .It corresponds to the Euler nutation in an elastic Earth.The increase in period from 305 days to 435 days is a consequence of the elastic yielding of the Earth.The superposition of the annual and Chandler frequencies results in a beat e ﬀect,in which the amplitude oftheFig. 2.16Variation of latitude due to superposition of the 435 day Chandler wobble period and an annualseasonal component (after Carter, 1989).latitude variation is modulated with a period of 6–7 years (Fig.2.16).2.3.4.4Precession and nutation of the rotation axisDuring its orbital motion around the Sun the Earth’s axis maintains an (almost) constant tilt of about 23.5Њto the pole to the ecliptic.The line of intersection of the plane of the ecliptic with the equatorial plane is called the line of equinoxes.Two times a year,when this line points directly at the Sun,day and night have equal duration over the entire globe.In the theory of the tides the unequal lunar attractions on the near and far side tidal bulges cause a torque about the rotation axis,which has a braking e ﬀect on the Earth’s rotation.The attractions of the Moon (and Sun) on the equatorial bulge due to rotational ﬂattening also produce torques on the spinning Earth.On the side of the Earth nearer to the Moon (or Sun) the gravitational attraction F 2on the equatorial bulge is greater than the force F 1on the distant side (Fig.2.17a).Due to the tilt of the rotation axis to the ecliptic plane (23.5Њ),the forces are notcollinear.A torque results,which acts about a line in the equatorial plane,normal to the Earth–Sun line and normal to the spin axis.The magnitude of the torque changes as the Earth orbits around the Sun.It is minimum (and zero) at the spring and autumn equinoxes and maximum at the summer and winter solstices.The response of a rotating system to an applied torque is to acquire an additional component of angular momentum parallel to the torque.In our example this will be perpendicular to the angular momentum (h ) of the spinning Earth.The torque has a component (t ) parallel to the line of equinoxes (Fig.2.17b) and a component normal to this line in the equatorial plane.The torque t causes an increment ⌬h in angular momentum and shifts the angular momentum vector to a new position.If this exercise is repeated incrementally,the rotation axis moves around the surface of a cone whose axis is the pole to the ecliptic (Fig.2.17a).The geographic pole P moves around a circle in the opposite sense from the Earth’s spin.This motion is called retrograde precession .It is not a steady motion,but pulsates in sympathy with the driving torque.A change in orientation of the rotation axis a ﬀects the location of the line of equinoxes and causes the timing of the equinoxes to change slowly.The rate of change is only 50.4 seconds of arc per year,but it has been recognized during centuries of observation.For example,the Earth’s rotation axis now points at Polaris in the constellation Ursa Minor,but in the time of the Egyptians around 3000BC the pole star was Alpha Draconis,the brightest star in the constellation Draco.Hipparchus is credited with dis-covering the precession of the equinoxes in 120 BC by comparing his own observations with those of earlier astronomers.The theory of the phenomenon is well understood.The Moon also exerts a torque on the spinning Earth and contributes to the precession of the rotation axis (and equinoxes).As in the theory of the tides,the small size of the Moon compared to the Sun is more than compen-sated by its nearness,so that the precessional contribution of the Moon is about double the e ﬀect of the Sun.The theory of precession shows that the period of 25,700yr is proportional to the Earth’s dynamical ellipticity ,H (see Eq.(2.45)).This ratio (equal to 1/305.457) is an impor-tant indicator of the internal distribution of mass in the Earth.The component of the torque in the equatorial plane adds an additional motion to the axis,called nutation ,because it causes the axis to nod up and down (Fig.2.17a).The solar torque causes a semi-annual nutation,the lunar torque a semi-monthly one.In fact the motion of the axis exhibits many forced nutations ,so-called because they respond to external torques.All are tiny per-turbations on the precessional motion,the largest having an amplitude of only about 9 seconds of arc and a period of 18.6yr.This nutation results from the fact that the plane of the lunar orbit is inclined at 5.145Њto the plane of the ecliptic and (like the motion ofartiﬁcial EarthFig. 2.17(a) The precession and forced nutation (greatly exaggerated)of the rotation axis due to the lunar torque on the spinning Earth (after Strahler, 1963). (b) Torque and incremental angular momentum changes resulting in precession.。

语法填空专项讲解+课件-2024届高三英语一轮复习

说明
shy → shyly dry → dryly
(7) ll结尾的，加y full →fully dull→dully
（8）ic 结尾的，加ally
basic → basically → scientifically
scientific
数词
num.
基数词指描述事物数量多少的词。
数词
序数词指表示顺序的数词。
elegant observe distance
although
sweep
sickness loose
because
sort
humorous
n.
adj.
v.
conj.
Part Of Speech
Part Of Speech
副词 adv.
形容词 adj.
名词 n.
六大实词
数词 num.
代词 pron.
is constructed
much
___68___ (construct)＂ Data about the moon’s composition, such as how ___69___ ice and other treasures it
its
contains, could help China decide whether ____70____ (it) plans for a future lunar (月球的) base are practical.
（表示无生命的东西的所有关系）
形容词副词
adj./adv.
形容词副词
比较级最高级
反义词
-(e)r/more-(e)st/most-
disun-

欧洲文化常识测试英语题型

《欧洲文化入门》复习题（一）Division One: Greek Culture and Roman CultureGreek CultureI.填空1.European culture is made up of many elements, two of these elements are consideredto be more enduring and they are the Greco-Roman element and the Judeo-Christian element.2.Greek culture reached a high point of development in the 5th century.3.In the second half of the 4th century B. C., all Greece was brought under the ruleof Alexander, king of Macedon.4.In 146 B. C. the Romans conquered Greece.5.Greek culture reached a high point of development in the 5th century.6.Revived in 1896, the Olympic Games have become the world’s foremost amateursports competition.7.Ancient Greeks considered Homer to be the author of their epics.8.The Iliad deals with the alliance of the states of the southern mainland of Greece,led by Agamemnon in their war against the city of Troy.9.The Odyssey deals with the return of Odysseus after the Trojan war to his home,island of Ithaca.10.Of the many lyric poets of ancient Greece, two are still admired by readers today:Sappho and Pindar.11.Sappho was considered the most important lyric poet of ancient Greece.12.Pindar is best known for his odes celebrating the victories at the athletic games,such as the 14 Olympic odes.13.The three great tragic dramatists of ancient Greece are Aeschylus, Sophocles,and Euripides.14.Aeschylus wrote such plays as Prometheus Bound, Persians and Agamemnon.15.Sophocles wrote such tragic plays as Oedipus the King, Electra, and Antigone.16.Euripides wrote mainly about women in such plays as Andromache, Medea, and TrojanWomen.edy also flourished in the 5th century B. C.. Its best writer was Aristophanes,who has left eleven plays, including Frogs, Clouds, Wasps and Birds.18.Herodotus is often called “Father of History”. He wrote about the wars be tweenGreeks and Persians.19.Thucydides described the war between Athens and Sparta and between Athens andSyracuse, a Greek state on the Island of Sicily.20.Pythagoras was a bold thinker who had the idea that all things were numbers.21.Pythagoras was the founder of scientific mathematics.22.Heracleitue believed fire to the primary element of the universe, out of whicheverything else had arisen.23.The greatest names in European philosophy are Socrates, Plato and Aristotle.24.Democritus was one of the earliest philosophical materialists and speculatedabout the atomic structure of matter.25.In the 4th century B. C., four schools of philosophers often argued with eachother, they arethe Cynics, the Sceptics, the Epicureans, and the Stoics.26.Euclid is well-known for his Elements, a textbook of geometry.27.To illustrate the principle of the level, Archimedes is said to have told theking: “Give me a place to stand, and I will move the world.”28.Greek architecture can be grouped into three styles: the Doric style which isalso called the masculine style; the Ionic style which is also called the feminine style; and a later style that is called the Corinthian style.29.The Acropolis at Athens and the Parthenon are the finest monument of Greekarchitecture and sculpture in more than 2000 years.30.In the 20th century, there are Homeric parallels in the Irishman James Joyce’smodernist masterpiece Ulysses.II.选择1.Which culture reached a high point of development in the 5th century B. C.?A.Greek CultureB.Roman CultureC.Egyptian CultureD.Chinese Culture2.In ___________ the Roman conquered Greece.A.1200B.C.B.700 B.C.C.146 B. C.D.The 5th century3.Which of the following works described the war led by Agamemnon against the cityof Troy?A.Oedipus the KingB.IliadC.OdysseyD.Antigone4.Which of the following is NOT the plays written by Aeschylus?A.AntigoneB.AgamemnonC.PersiansD.Prometheus Bound5.Which of the following is NOT the plays written by Sophocles?A.ElectraB.AntigoneC.Trojan WomanD.Oedipus the King6.Which of the following is the play written by Euripides?A.AntigoneB.PersiansC.ElectraD.Medea7.Which of the following is NOT the greatest tragic dramatist of ancient Greece?A.AristophanesB.EuripidesC.SophoclesD.Aeschylus8.Who ever said that “You can not step twice into the same river”?A.PythagorasB.HeracleitusC.Aristotle9.Who was the founder of scientific mathematics?A.HeracleitusB.AristotleC.SocratesD.Pythagoras10.Who is chiefly noted for his doctrine that “man is the measure of all things”?A.ProtagorasB.PythagorasC.PyrrhonD.EpicurusIII.名词解释1.Aeschylus2.Plato3.The CynicsIV.简答与问答1.What are the major elements in European culture?2.What were the main features of ancient Greek society?3.Who were the outstanding dramatists of ancient Greece? What important plays dideach of them write?4.Tell some of Plato’s ideas. Why do people call him an idealist?5.Give some examples to show the enormous influence of Greek culture on Englishliterature.Roman CultureI.填空1.The burning of Corinth in 146 B. C. marked Roman conquest of Greece, which wasthen reduced to a province of the Roman Empire.2.The Roman writer Horace said: “Captive Greece took her rude conqueror captive”.3.In 27 B. C. Octavius took supreme power as emperor with the title of Augustus.4.The Romans enjoyed a long period of peace lasting two hundred years, a remarkablephenomenon in history known as the Pax Romana.5.In the 4th century, the emperor Constantine moved the capital from Rome toByzantium, renamed it Constantinople ( modern Istanbul ).6.In 476 the last emperor of the west was deposed by the Coths and marked the endof the West Roman Empire.7.The East Roman Empire collapsed when Constantinople fell to the Turks in 1453.8.Julius Caesar recorded what he did and saw in the various military campaignshe took part in and these writings, collected in his Commentaries, are models of succinct Latin.9.Virgil was the greatest of Latin poets and wrote the great epic, the Aeneid.10.The Pantheon is the greatest and the best preserved Roman temple, which was builtin 27 B. C. And reconstructed in the 2th century A. D..11.She-wolf is the statue which illustrates the legend of creation of Roman.II.选择1.Who wrote, “I came, I saw, I conquered”?A.HoraceB.Julius CaesarC.VirgilD.Marcus Tullius Cicero2.The author of the philosophical poem On the Nature of things is ___________.A.VirgilB.Julius CaesarC.HoraceD.Lucretius3.Which of the following is not Roman architecture?A.The ColosseumB.The PanthenonC.The ParthenonD.Pont du Gard4.Who wrote, “Captive Greece took her rude conqueror captive”?A.SapphoB.PlatoC.VirgilD.HoraceIII.名词解释1.Julius Caesar2.The Pax RomanaIV.简答与问答1.What did the Romans have in common with the Greeks? And what was the chiefdifference between them?2.What is the book for which Virgil has been famous throughout the centuries? Inwhat way is the book linked with the Greek past?3.Why do we say Aeneas is a truly tragic hero?Division Two: The Bible and ChristianityThe Old TestamentⅠ填空题1.Among all the religions by which people seek to worship, Christianity is by farthe most influential in the West.2.Both Judaism and Christianity originated in Palestine the hub of migration andtrade routes, which led to exchange of ideas over wide areas.3.Some 3800 years ago the ancestors of the Jews –the Hebrews –wandered throughthe deserts of the Middle East.4.About 1300 B.C., the Hebrews came to settle in Palestine, known as Canaan atthat time, and formed small kingdoms.5.The king of the Hebrews was handed down orally from one generation to anotherin the form of folktales and stories, which were recorded later in the Old Testament.6.The Bible is a collection of religious writings comprising two parts: the OldTestament and the New Testament.7.The old Testament consists of 39 books, the oldest and most important of whichare first five books, called Pentateuch.8.When the Hebrews left the desert and entered the mountainous Sinai, Moses climbedto the top of the mountain to receive from God message, which came to be known as the Ten Commandments.9.Chronologically Amos is the earliest prophet in the Old Testament.10.In Babylon in the 6th century B.C., the Hebrews, now known as Jews, formedsynagogues to practise their religion.II 选择题1.Which of the following is by far the most influential in the West?_______A. BuddismB. IslamismC. ChristianityD. Judaism2.The Old Testament consists of 39 books, the oldest and most important of whichare the first five books, called __________.A. ExodusB. CommandmentsC. AmosD. Pentaeuch3.Which of the following is NOT the content of the Ten Commandments?_______A.Honour your father and your motherB.Do not commit suicideC.Do not desire your neighbour’s wifeD.Do not take the name of God in vain4.When in Babylon the Hebrews formed synagogues to practise their religion? ______A. in 169B.C. B. in the 4th centuryC. in 76 B.C.D. in the 6th centuryⅢ名词解释1.the Bible2.the Pentateuch3.Ten CommandmentsⅣ简答与问答1.What was the Hebrews major contribution to world civilization?2.Why do we say Judaism and Christianity are closely related?3.What are the Ten Commandments about?Rise of ChristianityⅠ填空题1.At the age of 30, Jesus received the baptism at the hands of John Baptist.2.Jesus spent most of his life in Galilee, where he apparently made a sensation.3.Jesus of Nazareth lived in Palestine during the reign of the first Roman EmperorAugustus.4.Jesus went with his disciples to Jerusalem for the Passover, but was betrayedby Juda.5.In 313 the Edict of Milan was issued by Constantine I and granted religiousfreedom to all and made Christianity legal.6.In 392 A.D, Emperor Theodosius made Christianity the official religions of theempire and outlawed all other religions.7.After Jesus died, St. Peter and St. Paul led the disciples of Jesus to spreadgospel in the Mediterranean regions.Ⅱ选择题1.After the _______ century Nestorianism reached China.A. sixthB. fifthC. secondD. third2.Which of the following emperors made Christianity the official religion of theempire and outlawed all other religions? __________A. TheodosiusB. AugustusC. Constantine ID. Nero Caesar3.Which of the following emperors issued the Edict of Milan and made Christianitylegal in 313? __________A. AugustusB. ThedosiusC. NeroD. Constantine I4.At the age of 30, Jesus Christ received the baptism at the hands of _________.A. St. PeterB. St. PaulC. John BaptistD. JohnWycliffⅢ名词解释1.The Edict of MilanⅣ简答与问答1.How did the relations between Christians and the Roman government change?The New TestamentⅠ填空题1.By 300 A.D. each local church was called a parish and had a full time leader2.Towards the end of he fourth century four accounts were accepted as part of theNew Testament, which tells the beginning of Christianity.3.When as Jesus’ mother Mary was espoused to Joseph, befor e they came together,she was found with child of the Holy Ghost4.Jesus went with his disciples to Jerusalem for the Passover, but was betrayedby Juda and caught at the Last Supper.Translations of the BibleⅠ填空题1.Except a few passages in the related Armaic dialect the Old Testament wasoriginally written in Hebrew. And the New Testament was originally written ina popular form of Greek.2.The oldest extant Greek translation of the Old Testament is known as theSeptuagint, as according the fictional letter of Aristeas, it was translated by 72 translators in 72 days.3.The most ancient extant Latin version of the whole Bible is the Vulgate edition,which was done in 384 –405 A.D. by St. Jerome in common people’s language.4.The first English version of whole Bible was translated from the Latin Vulgatein 1382 and was copied out by hand by the early group of reformers led by John Wycliff.5.The most important and influential of English Bible is the “Authorized” orKing James’ version, first published in 1611.Ⅱ选择题1.By 1693, the whole of the Bible had been translated in _________languages.A. 228B. 974C. 1202D. 1542.The oldest extant Greek translation of the Old Testament is known as ________.A. the Latin VulgateB. the AristeasC. the “Authorized”D. the Septuagint3.When printing was invented in the 1500’s, the _______ Bible was the firstcomplete work printed.A. EnglishB. LatinC. AramaicD. Hebrew4.When did the standard American edition of the Revised Version appear? _______A. 1885B. 1611C. 1901D. 1979Division Three: The Middle AgesManor and ChurchⅠ填空题1.In European history, the thousand year period following the fall of the WestRoman Empire in the fifth century is called the Middle Ages.2.Between the fifth and eleventh centuries, West Europe was the scene of frequentwars and invasions.3.The Middle Age is a period in which classical, Hebrew and Gothic heritage merged.4.Feudalism in Europe was mainly a system of land holding – a system of holdingland in exchange for military service.5.In 732 Charles Martel, a Frankish ruler gave his soldiers estates known as fiefsas a reward for their service.6.The center of medieval life under feudalism was the manor.7.By the 12th century manor houses came to be called castle, which were made ofstone and designed as fortress.8.As a knight, he was pledged to protect the weak, to fight for the church, tobe loyal to his lord and to respect women of noble birth. These rules were known as code of chivalry, from which the western idea of good manners developed.9.In the medieval days a knight was trained for war by fighting each other in mockbatters called tournaments.10.After 1054, the Church was divided into the Roman Catholic Church and the EasternOrthodox Church.11.The most important of all the leaders of Christian thought was Augustine of Hippowho lived in North Africa in the fifth century.12.Under feudalism, people of western Europe were mainly divided into three classes:clergy, lords and peasants.13.The Pope not only ruled Roman and parts of Italy as a king, he was also the headof all Christian churches in western Europe.14.In the Medieval times the Church set up a church court – the Inquisition tostamp out so-called heresy.15.One of the most important sacraments was Holy Communion, which was to remindpeople that Christ had died to redeem man.16.To express their religious feelings, many people in the Middle Ages went onjourneys to sacred places where early Christian leaders had lived. The most important of all was Jerusalem.17.With a return attack against the Moslems, the Western Christians launched aseries of holy wars called the Crusades.Ⅱ选择题1.In the later part of the 4th century, which of the following tribes swept intoEurope from central Asia, robbing and killing a large numbers of the half civilized Germanic tribes? ________A. the MongoliansB. the HunsC. the TurkishD. the Syrians2.The Middle Ages is also called the _________.A. “Age of Christianity”B. “Age of Literature”C. “Age of Holy Spirit”D. “Age of Faith”3.According to the code of chivalry, which of the following is not pledged to dofor a knight? _______A. To be loyal to his lordB. To fight for the churchC. To obey without question the orders of the abbotD. To respect women of noble birth4.In 732, who gave his soldiers estates known as fiefs as a reward for their service?_________A. Charles Martel, a Frankish rulerB. Charles I, a Turkish rulerC. Constantine I, a Frankish rulerD. St. Benedict, a Italian ruler5.When was the Church divided into the Roman Catholic Church and Eastern OrthodoxChurch?_________A. after 1066B. after 1296C. after 1054D. after4766.Which of the following about the knight or noble in the Middle Ages in WesternEurope is NOT true?____________A.Almost all nobles were knights in the Medieval days.B. A noble began his education as a page at the age of seven.C.As a knight, he was pledged to fight for the church.D.At about fourteen, the page became a knight.7.When was a noble crowned as a knight in the Middle Ages in Western Europe? _______A.At the age of 14.B.When he was taught to say his prayers, learned good manners and ran errandsfor the ladies.C.At a special ceremony known as dubbing.D.When he was pledged to fight for the church.8.Which of the following is NOT true about what the monks must do before enteringthe monastery according to the Benedictine Rule?A.They had to attend service 6 times during the day and once at midnight.B.They could promise to give up all their possession before entering themonastery.C.They were expected to work 5 hours a day in the fields surrounding themonastery.D.They had to obey without question the orders of the abbot.9.Under feudalism, what were the three classes of people of western Europe?________A. clergy, knights and serfsB. Pope, bishop and peasantsC. clergy, lords and peasantsD. knights, nobles and serfs10.By which year the Moslems had taken over the last Christian stronghold and wonthe crusades and ruled all the territory in Palestine that the crusaders had fought to control? ________A. 1270B. 1254C. 1096D. 1291Ⅲ名词解释1.the Middle Ages2.Manor3.Code of Chivalry4.Benedictine Rule5.the CrusadesⅣ简答与问答1.Who was Charles Martel?2.What was the difference between a serf and a free man?3.Into what three groups were people divided under feudalism?4.What happened in Western Europe after the decline of the Roman Empire?Learning and Science, Literature, Art and ArchitectureⅠ填空题1.Charlemagne, who temporarily restored order in western and central Europe, wasperhaps the most important figure of the medieval period.2.Charlemagne was crowed “Emperor of the Romans” by the Pope in 800.3.The Summa Theologica by St. Thomas Aquinas forms an enormous system and sumsup all the knowledge of medieval theology.4.Roger Bacon was one of the earliest advocates of Scientific research and calledfor careful observation and experimentation.5.“National epic” refers to the epic written in vernacular languages – thatis, the languages of various national states that came into being in the Middle Ages.6.Beowulf is an Anglo-Sexon epic, in alliterative verse, originating from thecollective efforts of oral literature.7.Dante Alighieri was the greatest poet of Italy, his masterpiece, The DivineComedy, is one of the landmarks of world literature.8.Chaucer was a great English poet, The Canterbury Tales were his most popularwork for their power of observation, piercing irony, sense of humor and warm humanity.9.Chaucer writers in dialect used by Londoners, and by the sheer weight andpopularity of his writings he sets it firmly on the way towards Modern English.10.The style of architecture under Romanesque art is characterized by massiveness,solidity and monumentality with all overall blocky appearance.11.The Gothic style started in France and quickly spread through all parts of westernEurope.Ⅱ选择题1.Which of the following was crowned “Emperor of the Romans” by the Pope in 800?______A. St. Thomas AquinasB. CharlemagneC. ConstantineD. King James2.Who was the ruler of the Anglo-Saxon kingdom of Wessex and contributed greatlyto the medieval European culture? _________A. Charles IB. Constantine IC. Alfred the GreatD. Charles the Great3.Does Song of Roland belong to which co untry’s epic? _________A. EnglishB. GermanicC. HebrewD. French4.Who is the author of the Opus Maius? ________A. Roger BaconB. Dante AlighieriC. ChaucerD. St. Thomas AquinasⅢ名词解释1.Carolingian Renaissance2.Beowulf3.Song of Roland4.The Canterbury tales5.Romanesque6.GothicⅣ简答与问答1.What was the merit which Charlemagne and Alfred the Great share?Division IV: Renaissance and ReformationRenaissance in ItalyⅠ填空题1.Generally speaking, Renaissance refers to the period between the 14th and mid17th century.2.Humanism is the essence of the Renaissance.3.In essence, Renaissance was a historical period in which the European humanistthinkers and scholars made attempts to get rid of conservatism in feudalist Europe and introduce new ideas that expressed the interests of bourgeoisie, to lift the restrictions in all areas placed by the Roman Church authorities. 4.Renaissance started in Florence and Venice with the flowering of paintings,sculpture and architecture.5.Beginning from the 11th century, cities began to rise in central and north Italy.6.Decameron is a collection of 100 tales told by 7 young ladies and 3 youngergentlemen on their way to escape the Black Death of 1348.7.Petrach was best known for Canzoniers, a book of lyrical songs written in hisItalian dialect.8.The Renaissance artists introduced in their works scientific theories of anatomyand perspective.9.The four representative artists of High Renaissance in Italy are Leonardo daVinci, michelangelo, Raphael and Titian.10.Loenardo da Vinci’s major works: Last Supper is the most famous of religiouspictures; Mona Lisa probably is the world’s most famous portrait.11.Michelangelo created a style of art in which he freed himself from the oldtradition of decoration on the one hand and documentary realism on the other.12.Titian’s painting is acknowledged to have established oil colour on canvas asthe typical medium of the pictorial tradition in western art.13.In world trade, Italy had lost its supremacy because of the discovery of Americain 1492 and the rounding of the Cape of Good Hope in 1488, the opening of an all-water route to India which provided a cheaper means of transport.14.Petrach is looked up as the father of modern poetry.15.Italy is regarded as the birthplace of the Renaissance.Ⅱ选择题1.Where did the Renaissance start with the flowering of paintings, sculpture andarchitecture? _______A. in Greece and RomeB. in Florence and VeniceC. in Milan and FlorenceD. in Italy and Germany2.When did the Renaissance reach its height with its center moving to Milan, thento Rome, and created High Renaissance? ___________A. in the 11th centuryB. in the 15th centuryC. in the 16th centuryD. in the 17th century3.Which of the following works is written by Boccaccio? _______A. DecameronB. CanzoniersC. DavidD. Moses4.Who is the author of the painting, Betrayal of Judas? ________A. GiottoB. BrunelleschiC. DonatelloD. Giorgione5.Which of the following High Renaissance artists is the father of the modern modeof painting? _______A. RaphaelB. TitianC. da VinciD.Michelangelo6.Which of the following High Renaissance artists was best known for his Madona(Virgin Mary)?A. TitianB. da VinciC. MichelangeloD. Raphael7.Which of the following paintings was based on the story in the Bible with Mariariding on a donkey ready to face the hardship ahead? ________A. TempestaB. Sacred and Profane LoveC. Flight into EgyptD. The Return of the HuntersⅢ名词解释1.Renaissance2.DecameronⅣ简答与问答1.What made Italy the birthplace of the Renaissance?2.What are the main elements of humanism? How are these elements reflected in artand literature during the Italian Renaissance?3.How did Italian Renaissance art and architecture break away from medievaltradition?4.In what way was Leonardo da Vinci important during the Renaissance?Reformation and Counter-ReformationⅠ填空题1.The Reformation led by Martin Luther which swept over the whole of Europe wasaimed at opposing the absolute authority of the Roman Catholic Church and replacing it with the absolute authority of the Bible.2.Martin Luther was the German leader of the Protestant Reformation. His doctrinemarked the first break in the unity of the Catholic Church.3.When the Pope refused to recognized Henry’s mar riage with Anne Boleyn, BritishParliament, in 1534, passed the Act of Supremacy which marked the formal break of the British with the Papal authorities.4.Ignatius and his followers called themselves the Jesuits, members of the Societyof Jesus.5.John Calvin put his theological thoughts in his Institutes of the ChristianReligion, which was considered one of the most influential theological works of all times.Ⅱ选择题1.Who took up the translation of the Bible into English for the first time? ________A. Jan HusB. John WyliffC. Martin LutherD. John Calvin2.Who is the author Institutes of the Christian Religion?A. John WycliffB. Jan HusC. John CalvinD.Erasmus3.In whose reign did the formal break of the British with the papal authorities take place?____A. Elizabeth IB. William IC. Edward IIID. Henry VIII4.After the formal break of the British with the papal authorities, who was the head of the church? _______A. KingB. PopeC. BishopD. QueenⅢ名词解释1.Calvinism2.the Council of Trent3.Counter-ReformationⅣ简答与问答1.What are the doctrines of Martin Luther?2.What was the significance of the Reformation in European civilization?Renaissance in other CountriesⅠ填空题1.The Protestant group in France was known as the Huguenots whose rivalry withthe Catholic Church led to the wars of religion from 1562 to 1598.2.In 1492 the Moors that had ruled Spain for four centuries were driven out fromtheir last stronghold.3.In 1492 Columbus discovered American and claimed America for Spain.4.The author of Don Quixote is Cervantes.5.Albrecht Dürer was the leader of the Renaissance in Germany. His engravings areunsurpassed and his paintings of animals and plants are exceedingly sensitive.6.Under the reign of Elizabeth I, England began to embark on the road tocolonization and foreign control that was to take it onto its heyday of capitalist development.7.Thomas More was a great humanist during the Renaissance. Among his writings thebest known is Utopia.8.Cervantes crowned literature of Spain and Shakespeare of England during theRenaissance.Ⅱ选择题1.Which of the following works was written by Rabelais, in which he praises thegreatness of man, expresses his love of love and his reverence and sympathy for humanist learning? _______A.Gargantua and PantagruelB. Don QuixoteC. The Praise of FollyD. Utopia2.Whose motto put down in his essays “What do Know” is world famous?________A. CervantesB. RabelaisC. MontaigneD. Shakespeare3.Which of the following works is worth reading for Montaigne’s humanist ideasand a style which is easy and familiar? ________A. SonnetsB. DecameronC. RabelaisD. Of Repentance4.Which of the following is NOT French writer poet? _______A. CervantesB. Pierre de RonsardC. RabelaisD. Montaigne5.In 1516 who published the first Greek edition of the New Testament?_________A. BruegelB. ErasmusC. El GrecoD. Rabelais6.“To be, or not to be, -- that is the question ” from whose works? _______A. ChaucerB. DanteC. Roger BaconD. ShakespeareⅢ简答与问答1.Why did England come later than other countries during the Renaissance? In whatway was English Renaissance different from that of other countries? Who were the major figures and what were their contributions?Science and Technology during the RenaissanceⅠ填空题1.The Renaissance was the golden age of geographical discoveries: by the year of1600 the surface of the known earth was doubled.2.Columbus was a Genoese-born navigator and discoverer of the New World.3.Dias was a Portuguese navigator who discovered the Cape of Good Hope.4.Vasco da Gama was a Portuguese navigator, who discovered the route to India roundthe Cape of Good Hope between the year of 1497 and 1498.5.Amerigo Vespucci was the Italian navigator in whose honor America was named6.Amerigo Vespucci discovered and explored the mouth of the Amazon and acceptedSouth America as a new continent.7.Copernicus came to be known as father of modern astronomy.8.During his life time Leonardo da Vinci dissected more than thirty corpse and。

拉马努金恒等式的证明

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Dynamics of Gravity in a Higgs Phase

a rXiv:h ep-ph/57120v221J ul26HUTP-05/A0030UTAP-530RESCEU-10/05Dynamics of Gravity in a Higgs Phase Nima Arkani-Hamed a ,Hsin-Chia Cheng a ,b ,Markus A.Luty a ,c ,d ,Shinji Mukohyama a ,e ,Toby Wiseman a a Jeﬀerson Laboratory of Physics,Harvard University Cambridge,Massachusetts 02138b Department of Physics,University of California Davis,California 95616c Physics Department,Boston University Boston,Massachusetts 02215d Physics Department,University of Maryland College Park,Maryland 20742∗e Department of Physics and Research Center for the Early Universe The University of Tokyo,Tokyo 113-0033,Japan Abstract We investigate the universal low-energy dynamics of the simplest Higgs phase for gravity,‘ghost condensation.’We show that the nonlinear dynam-ics of the ‘ghostone’ﬁeld dominate for all interesting gravitational sources.Away from caustic singularities,the dynamics is equivalent to the irrota-tional ﬂow of a perfect ﬂuid with equation of state p ∝ρ2,where the ﬂuid particles can have negative mass.We argue that this theory is free fromcatastrophic instabilities due to growing modes,even though the null energy condition is violated.Numerical simulations show that solutions generally have singularities in which negative energy regions shrink to zero size.We exhibit partial UV completions of the theory in which these singularities are smoothly resolved,so this does not signal any inconsistency in the eﬀective theory.We also consider the bounds on the symmetry breaking scale M in this theory.We argue that the nonlinear dynamics cuts oﬀthe Jeans instability of the linear theory,and allows M <∼100GeV.1IntroductionIs general relativity the correct description of gravity at long distances and times? Certainly,there are good reasons for thinking that this is the case.Experimentally, gravity has been probed at distance scales ranging from10−1mm(in short-range force experiments)to at least1014cm(the size of the solar system).Theoretically, general relativity is the unique Lorentz-invariant theory of massless spin2,and its conceptual elegance is beyond question.However,the situation is far less clear on cosmological distance and time scales.Structure formation,galaxy rotation curves and gravitational lensing,and the accelerating expansion of the universe cannot be explained by general relativity coupled to known matter.These anomalous eﬀects are conventionally attributed to‘dark matter’and‘dark energy.’However,given the fact that the observed eﬀects are purely gravitational,it makes sense to ask whether they may have a common origin in a modiﬁcation of gravity in the infrared.These considerations have led to a revival of interest in consistent infrared modiﬁcations of gravity[1,2,3,4,5,6,7].In the present paper,we further investigate the model of Ref.[5],‘ghost conden-sation.’This can be viewed as the universal low-energy dynamics associated with the simplest Higgs phase for gravity,arising when Lorentz symmetry is broken sponta-neously.Breaking of Lorentz symmetry is of course ubiquitous.For example,time-dependentﬁelds in cosmology deﬁne a preferred frame.However,these solutions are not the ground state of the theory:they carry energy density and dilute away as the universe expands.Any‘modiﬁcation of gravity’induced by such solutions becomes relevant only at scales of order the horizon.We are instead interested in the situation where Lorentz symmetry is broken inﬂat spacetime,allowing nontrivial modiﬁcation of gravity inside the horizon.This means that the symmetry breaking sector has peculiar properties;in particular,the stress-energy tensor must vanish in the ground state:Tµν =0.(1.1) Spontaneous breaking of Lorentz symmetry gives rise to a gapless scalar excitation analogous to the Nambu-Goldstone bosons that arise from spontaneous breaking of internal symmetries.‘Ghost condensation’gives rise to a single such mode,and is in this sense the minimal model of spontaneous breaking of Lorentz symmetry.We refer to the scalar mode as a‘ghostone boson.’In analogy with the Higgs phase for gauge theory,the ghostone mode mixes with the graviton,modifying gravity in a nontrivial manner.Ref.[5]studied this theory and analyzed the modiﬁcation of gravity in the weak-ﬁeld limit.The dynamics is governed by a consistent eﬀective theory deﬁned by the scale M where the symmetry is broken.It was shown that the ghostone mode gives a possible new origin for dark energy and dark matter.Ref.[8]showed that the ghostone mode may also be the inﬂaton,leading to interesting testable consequences.In the present paper,we further investigate the dynamics of this theory.We show that nonlinearities dominate the dynamics of the ghostone sector for all gravitational sources of interest.In particular,the time scale for the onset of nonlinear dynamics for aﬁxed gravitational source is precisely the infall(or orbit)time associated with the source.Away from singularities,the nonlinear solutions are equivalent to the gradientﬂow of aﬂuid with equation of stateρ2p=3,we discuss the nonlinear dynamics of the ghost condensate.We discuss the time scales and present theﬂuid picture.In section4,we show the numerical simulations of the nonlinear evolutions and discuss the resolution of caustic singularities.In section 5,we discuss the bounds of this theory.This includes mass and energy accretion in slow-moving objects,gravitational lensing and energy loss from moving objects. We do not claim to have a complete understanding of the dynamics,so this section is intended to be preliminary and provocative.In section6,we brieﬂy discuss the possibility that ghost condensate may constitute the dark matter.We show that the initial growth of the density perturbations in the linear regime is identical to that of the standard cold dark matter.Whether it can form the correct structure depends on the details of the nonlinear evolution which is left for future investigations.Our conclusions are presented in section7.2Review of the Linear Theory2.1Eﬀective TheoryWhat is a Higgs phase for gravity?It is easiest to answer this question in linearized general relativity,where we expand the metric aboutﬂat spacegµν=ηµν+hµν(2.1)and keep only terms quadratic in hµν.In this theory,theﬁelds hµνare closely analo-gous to gaugeﬁelds with gauge transformation lawδhµν=−(∂µξν+∂νξµ),(2.2)whereξµare the generators of inﬁnitesmal diﬀeomorphismsxµ→xµ+ξµ.(2.3) We want to consider the case where the time diﬀeomorphisms generated byξ0are spontaneously broken.This means that time diﬀeomorphisms are realized nonlinearly in the eﬀective theory containing only the ghostoneﬁeld.The minimal model contains a single real ghostoneﬁeldπthat shifts under time diﬀeomorphisms:δπ=−ξ0.(2.4)Note thatπnaturally has units of time.We now write the most general eﬀective Lagrangian invariant under these symmetries.This contains the Einstein Lagrangian,and additional terms constructed from the invariantsΣ=˙π−1(˙h ij−∂i h0j−∂j h0i+2∂i∂jπ).(2.6)2The leading terms areL eﬀ=L E+M4 12h00 2−α12M2K2+O(π3) .(2.7) (Note that we have built in the fact thatﬂat space is a solution by not writing any linear terms in the Lagrangian.)In the limit where we turn oﬀgravity,we see that the ghostone mode has dispersion relationα k4ω2=A conventional eﬀective Lagrangian for this theory isL=+1φ=0.This has solutions withφ=nµxµfor any constant 4-vector nµ.If nµis timelike,we can choose the time direction so that the solution isφ=ct.(2.11)This is a solution for any constant c.Atﬁrst sight it may appear that these are obviously not candidate ground states of the theory,but the situation is actually more subtle.Suppose we expand inﬂuctuations about this solutionφ=ct+π.(2.12)Note that under time diﬀeomorphisms,πtransforms asδπ=−ξ0/c,(2.13)just like the ghostone mode considered above.Expanding the Lagrangian to quadratic order inπ,oneﬁnds that theﬂuctuations forπhave good time and space kinetic terms.This means that the solution is stable under localﬂuctuations for any value of c!The reason for this is that the theory has a conserved current associated with the shift symmetryJµ=∂µφ.(2.14)Solutions with c=0have a constant nonzero charge density.Local excitations cannot change the total charge,so conﬁgurations with lower energy cannot be reached. However,when we turn on gravity,solutions with c=0will cause the universe to expand,and the charge will dilute away.Lorentz invariance is therefore not broken spontaneously in this theory.Consider instead an eﬀective Lagrangian of the formL eﬀ=M4P(X),X=∂µφ∂µφ.(2.15) Note thatφhas dimensions of length(or time),so that X is dimensionless.This omits only terms with more than one derivative acting onφ,such as(We see that small perturbations are stable provided2c2P′′(c2)+P′(c2)>0,P′(c2)>0.(2.17) (Note that a conventional kinetic term P(X)=+12(X−1).(2.22)Stability in the linearized theory requires higher-derivative terms to give a nonzero spatial kinetic term,for example∆L eﬀ=−αM2φ)2=−αM2αM2M2−,T J∼M2Pl M2ScreeningExponentially fallingPotentialFig.2.A schematic illustration of the screening eﬀect for gauge theories in the Higgs phase.For M<∼10MeV,T J is longer than the lifetime of the universe and there is clearly no constraint from the Jeans instability.2.4Negative EnergyEven for time shorter than the Jeans time scale T J,stability is an issue because the ghostone energy can be negative.Recall that the ground state X=1is the boundary of the stability region.Expanding to higher orders inπ,weﬁnd interaction terms such asL eﬀ=M4 12˙π( ∇π)2+···−α˙π 1/2.(2.27) Quantum-mechanically,there areﬂuctuations at all length scales.Nonetheless,Ref.[5] showed that there is no quantum instability in the eﬀective theory using a scaling ar-gument.If we scale energy by E→sE,the quadratic kinetic terms are left invariant−1Anti−ScreeningOscillating PotentialFig.3.A schematic illustration of the anti-screening eﬀect for gravity in the Higgs phaseif we scalet→s−1t,(2.28)x→s−1/2 x,π→s1/4π.With this scaling the cubic operator˙π( ∇π)2in Eq.(2.26)scales as s1/4and is therefore (barely!)irrelevant.All other operators are even more irrelevant,showing that there is a regime of low energies where the expansion is under control.2.5Preview of Nonlinear EﬀectsThe arguments above show that the eﬀects of the nonlinear terms are under control at low energies and smallﬁeld amplitudes.However,in the presence of large classical gravitational sources the nonlinear terms can become important.In fact,the time scale for the ghostoneﬁeld near a classical source is just the gravitational infall time of the source.This can be understood from the form of the stress-energy tensor.In the approximation where the Lagrangian is L=P(X),the stress-energy tensor has the formTµν∝−P(X)gµν+2P′(X)uµuν.(2.29)whereuµ=∂µφ.(2.30) This has the form of a stress-energy tensor for a perfectﬂuid with4-velocity uµ.1 The fact that uµis a gradient means that theﬂow is irrotational:∂[µuν]=0implies ∇× u=0.The equations of motion for the ghostoneﬁeld follow from the conservation of the stress-energy tensor∇µTµν=0,which are intepreted as the conservation of energy and momentum in theﬂuid.In the presence of a classical gravitational source, aﬂuid clearly responds on a time scale given by the infall time,and therefore so does the ghostone mode.As it will be shown in the next section,this is exactly the time scale where the nonlinear term becomes important.For smallﬂuctuations about the minimum X=1,P(X)can be approximated byP(X)≈12Σ2,(2.31)where an overall contribution to the cosmological constant has been removed.We can read oﬀthe equation of state of theﬂuid from the energy-momentum tensor.It isp=ρ21The deﬁning property of a perfectﬂuid is that at each point there is a frame in which the stress-energy tensor has the form Tµν=diag(−ρ,p,p,p).Such phenomena were also observed in some other scalarﬁeld theories[11].Near these singularities,the higher-derivativeα( ∇2π)2term can no longer be neglected. This term contributes positive gradient energy,and weﬁnd in numerical simulations that it generically smoothly resolves the caustic singularity,giving rise to a‘bounce’with outgoingπwaves and positive and negative regions ofΣnear the would-be caustic.The fact thatΣ(and henceρ)can be negative brings up again the question of the stability of the theory.As discussed above,regions withΣ<0modes with suﬃciently long wavelengths are unstable.In numerical simulations,weﬁnd that negative energy regions tend to shrink while the amplitude ofΣgrows inside the region.This can be understood from theﬂuid picture,since this is valid in the limit where we neglect the( ∇2π)2term,which is a good approximation away from caustic singularities.In this picture theΣ<0region consists ofﬂuid particles with negative mass.It is therefore clear that energy(mass)cannotﬂow across the boundary of theΣ<0region,since the boundary consists of particles with vanishing mass.The boundary can move however,and in theΣ<0region the positive pressure favors large gradients and causes theΣ<0region to shrink.Numerical simulations show that someΣ<0regions continue to shrink until they exit the regime of validity of the eﬀective theory.These singularties need to be resolved in a more fundamental theory.Similar conclusion was also obtained in Ref.[12]which studied the two-dimensional case.However,we show that the total energy inside theΣ<0regions formed in astrophysical situations is very small,and does not lead to any observable consequences provided the singularities are regulated in a smooth way.We will discuss a partial UV completion to do this,and present numerical evidence that it works.The nonlinear dynamics aﬀects the bounds on M derived in the linear theory. The Jeans instability in the linear theory gives a bound M<∼10MeV if we require that there is no exponential growth of the oscillatory potential within the age of the universe.However,the nonlinear dynamics is expected to cut oﬀthe instability, and may weaken this bound.We argue below that the strongest bound comes from gravitational lensing due to regions of positive and negative energy produced by the Jeans instability.Demanding that the random walk of light rays due to the lensing does not smear out the observed CMB anisotropies gives the bound M<∼100GeV.We also consider other possible bounds on the ghost condensate from the grav-itational sector.The modiﬁcations of the gravitational potential are small due to velocity eﬀects[13,14].We also consider energy loss in the nonlinear theory,as well as energy stored in would-be caustic singularities.Weﬁnd that these eﬀects are negligible,and we believe that the theory is safe for M<∼100GeV.3Nonlinear DynamicsWe now turn to the nonlinear dynamics of the theory.The nonlinear dynamics is veryrich and complex,and we emphasize that we do not claim a complete understandingin this work.It is therefore important to keep in mind that there is a simple limit ofthis theory,independent of the details of the nonlinear dynamics[5].The Ghostonesector naturally couples to matter only through gravity.(Gravitationally induceddirect couplings to standard modelﬁelds are easily seen to be negligibly small.)Themaximum value of the gravitational energy in the Ghostone energy is of order M4,which does not aﬀect even cosmology if M<∼(M Pl H0)1/2∼10−3eV.Such low values of M are still very interesting for cosmology,since the ghost may be a source of darkenergy and dark matter[5]and may drive inﬂation[8].Another general point to keep in mind in the following is that the modiﬁcationsof gravity vanish in regions whereΣ=0(X=1)and we neglect the( ∇2π)2term. This is because in this limit,the Lagrangian in‘unitary gauge’φ=t(φ=0)isL=√where∇2Φ=T00(X−1)=˙π−12Σ2−α2π2T NL∼√we have T2NL∼r3/2/R S,which is the Kepler relation.This is a very direct way of seeing that the nonlinear eﬀects become important on the gravitational time scale.Solving Eq.(3.7)forπ,we obtain| ∇π|∼πΦ≪1.(3.9) That is,ghostone amplitudes are small for weak gravity,in agreement with our as-sumptions.On the other hand,in the linear approximation theα( ∇2π)2becomes comparable to˙π2at T Lin∼ML2.As a result,the nonlinear evolutions completely dominates for T NL<∼T Lin∼ML2,orΦL2>∼1M src M Pl(X−1)2=18Σ2gµν+Σuµuν ,(3.13)2whereuµ=∂µφ.(3.14) Note that uµis nonzero and timelike everywhere.This has the form of the stress-energy tensor for a perfectﬂuid with4-velocity uµ.Because uµis the gradient of a scalar,theﬂow of theﬂuid is irrotational.The conservation of the stess-energy tensor∇µTµν=0gives the equation of motion for the ghostoneﬁeld,and also gives the Euler equation for theﬂuid.ForΣ≪1we can read oﬀthe density and pressure.(3.15)ρ=M4Σ,p=12M4This establishes the equivalence of the ghostone theory without theα( ∇2π)2term to the irrotationalﬂow of a perfectﬂuid2.It is also insightful to understand the equivalence directly in terms of the equa-tions of motion.For the ghostoneﬁeld,the equations of motion have the form of a conservation law˙Σ= ∇·[Σ ∇π],(3.16) whereΣis the charge density.For aﬂuid made of particles carrying the conserved charge,the current is J=Σ v,so we identifyv=− ∇π.(3.17) Again we see that theﬂuidﬂow is irrotational.In thisﬂuid picture,the equations of motion are satisﬁed simply due to the fact that theﬂuid particles carry their charges with them.It remains only to satisfy the relation betweenΣandπ:Σ=˙π−1=− ∇(Φ+Σ),(3.19)Dt4πG Nρcorresponding to this equation of state is˜L J∼M P l/M2, where c s= ρ/M4is the sound velocity.Intriguingly,this agrees with the Jeans length (2.25)in the linear theory up to a constant of order unity.This equation of state ignores the k4 term,while the linear analysis does not take into account the nonlinear termΣ2in p.Hence,it is not a priori clear whether these two Jeans length should be the same or not.Nonetheless,they agree.whereD+ v· ∇(3.20)∂tis the time derivative along the particle worldline(also called the convective or Lagran-gian derivative).Eq.(3.19)is just Newton’s law for a particle moving in a potential Φ+Σ.Using the identiﬁcations Eq.(3.15)we can write− ∇Σ=−13Note that the equivalence between the ghostone andﬂuid pictures is a kind of duality,since it exchanges a constraint equation with an equation of motion.Noether charge of the original time translation symmetry,which we refer to as the ‘gravitational energy.’This is not the same as the Noether charge associated with the time translation symmetry that is unbroken in the vacuum,which we call the ‘inertial energy.’The inertial energy εis the energy associated with the unbroken time translation symmetry of the Lagrangian Eq.(3.6).(It is also the Hamiltonian density of the system.)Conservation of inertial energy states that˙ε=− ∇· p ,(3.22)whereε=M 4 12Σ( ∇π)2+αM 2( ∇2π) ∇π−˙π ∇( ∇2π) (3.24)is the momentum density.In the linearized approximation,ε=M 4 12M 2( ∇2π)2 ≥0.(3.25)However,in the nonlinear theory the inertial energy is not positive deﬁnite due to the second term in Eq.(3.23).The energy density can be negative only in regions where Σ<0.In fact it is easy to see that the energy is unbounded from below.For example,for π=c | x |we have ε=−1M 2 ∇( ∇2π).(3.27)If we neglect theα( ∇2π)2term,this conservation law is taken into account very directly in theﬂuid picture,to which we now turn.In this approximation,negative energy regions correspond precisely to regions whereΣ<0,so we consider such a region surrounded byΣ>0.5In theﬂuid picture,the conserved charge is carried by the individual particles,so theΣ<0region consists of particles with negative charge,while the boundary of theΣ<0region consists of particles of vanishing charge.Therefore,there can be noﬂux of charge across theΣ=0boundary,and the total charge inside the region does not change.On the other hand,theΣ=0boundary can move.Since ∇Σpoints outward at the boundary,the equation of motion forﬂuid particles Eq.(3.19)implies that the particles on the boundary experience an inward force due to the pressure.Therefore, theΣ<0region tends to shrink.These arguments show that the total shift charge integrated over aΣ<0regionQ= Σ<0d3xΣ(3.28)does not change with time.This can also be seen from the fact that the shift current J vanishes on theΣ=0boundary.The shift charge is not the same as the total inertial energyE= Σ<0d3xε.(3.29)However,if we neglect theα( ∇2π)2term,theﬂux of inertial energy across theΣ=0 boundary also vanishes,and therefore E also does not change with time.As with theﬂuid picture,these results hold beyond the approximations made here.This is discussed in the appendix.3.5Caustic SolutionsTheﬂuid picture can be used to understand the structure of the caustic singularties that occur when we neglect theα( ∇2π)2term.We restrict attention to the case Σ=0,where there is no pressure and theﬂuid particles follow geodesics.In this case,it is clear that there are caustics without theα( ∇2π)2term.It is possible that the pressure resolves the caustics in important situations such as inside galaxy halos made of ghostone dark matter,but we will not consider that here.To introduce the subject of caustics,we consider the gravitational potential due to a uniform sphere of matter with densityρ0.Inside the sphere,the gravitational potential isρ0Φ=,(3.33)Twhere T is a constant that tells us how the initial velocity varies away from x0=0. Solving Eq.(3.31)for x0we obtainxx0=−TFig.4.The1-dimensional‘perfect caustic.’The shaded regions shows the region where we can expand the solution perturbatively about the perfect caustic.Since the velocity is constant along any trajectory,we havexv(x)=v0(x0)=−.(3.36)2(t−T)Note that this solution is scale-free,since T just gives the time to the caustic.Shifting t→t−T,we obtainx2π(x,t)=−=0(3.38)∂x0forﬁxed t.This gives the time to the caustic for a given value of x0as1t c(x0)=−In a realistic case,the function v0(x0)will be more complicated.We want to expand about the point where the causticﬁrst forms,i.e.the minimum of t c(x0), which we take to be x0=0.Expanded about this point,v0takes the formv0(x0)=−x06L2Tx30+O(L−3),(3.40)where we have performed a boost so that v0(0)=0.Because of the focussing of the geodesics,this expansion is valid only for|x|≪L t−Tt−T−T3(t−T)4+O(L−3).(3.42)and thereforev(x)=x6L2x3L21We can use Eq.(3.44)to estimate the time and distance scale where theα( ∇2π)2 term becomes important.This will happen whenα∂4xπML2 1/2,(3.46)where∆t=T−t is the time to the caustic.The distance scale where theα( ∇2π)2 term becomes important is therefore∆x∼LM 1/2.(3.47)Note that∆x,∆t≫M−1as long as L,T≫M−1and L/T≪1(i.e.the system is nonrelativistic),so this is within the regime of validity of the eﬀective theory.4Numerical SimulationsIn this section we describe various numerical simulations of the ghostﬁeld which allow us to understand the rather exotic features of its non-linear evolution.The test cases weﬁrst present assume symmetry to reduce the dynamics to a one dimensional problem,and will see that the various symmetries have diﬀerent be-haviours.We will discuss how caustics do indeed form in the theory.As mentioned earlier,this is clear forα=0,but one would naively expect non-zeroαto amelio-rate this problem.However,as we will show,this is not the case,and for certain symmetries the‘perfect’caustic remains an attractor.As one might expect,the singular behaviour becomes less strong as one moves from planar through axisymmetry to spherical symmetry.Indeed without any grav-itational potential we will see the planar reduction exhibits singularities,while the spherical theory without potential does not.However,once a gravitationally at-tractive potential is added,all three symmetries become singular under evolution of regular initial data.Clearly assuming symmetry can lead to unphysical behaviours,whilst the phy-sical situation we are ultimately interested,namely structure formation,is expectedto have very little symmetry.Hence this section will conclude with a study of a3-d numerical evolution where no continuous symmetry is present and a moving gravi-tational potential seeds the ghostﬁeld growth.As expected from the1-d examples, we againﬁnd singular caustics do develop,and interestingly appear to take a planar form.4.1One dimensional evolutions:Planar,axial and spherical symmetryWe will initially consider the case of evolution of the ghostﬁeld in the absence of gravitational sources,seeded instead from a local perturbation in the ghostﬁeld itself.Then reducing to planar symmetryπ=π(t,r)and we may write the equation for the ghost decoupled from gravity in a manifestlyﬂux conservative form,˙H=∂r Σ+1α/M,the second line is simply the deﬁnition ofΣand˙x,x′are the time and space derivatives of x.We then use a Crank-Nicholson method to evolve our initial data,which we take to be a Gaussian proﬁle inππ(t=0)=π0e−r2(4.2) withΣ=0initially.As discussed previously,with L=0Σwould remain zero,but the higher derivative term sourcesΣ.In order to make contact with the epoch of structure formation we wish to have moderate initial amplitudes so|π0|<1but is still of order unity.From(4.2)we have chosen units to have initial data with unit spatial variation.We then wish to have L,the ghost length scale,to be L≪1.Naturally numerical methods limit our ability to separate the initial data length scale from the ghost length whilst maintaining accuracy.However we may separate the scale suﬃciently to see the asymptotic properties of taking L very small.Theﬁrstﬁgures we show,5and6,illustrate the generic behaviour for the planar system for small L,showing both H andΣfor the evolution.We show both the evolution of data for L=0.005and also for L=0.for comparison,and the initial amplitude was taken to beπ0=0.1.As expected L=0.evolution leads to a caustic,whose time for formation scales as t caustic≃1/π0.Adding the higher derivative term changes the evolution dramatically around this time scale,completely smoothing out the ghostﬁeld.However,we seefrom theﬁgure that the radiated waves scattered by the action of this higher derivative term in fact are attracted back to the origin where they grow and become singular. Obviously since the symmetry is planar,this growth is not due to a measure factor,but rather is due a‘perfect caustic’forming,which as mentioned before cannot be rescued by the higher derivative term we use here.Indeed this perfect caustic formation can be seen locally in detail fromΣnear the singularity.We note that taking suﬃciently large L∼√π0when T→∞),or with decreasing L(so that the radiation from the increasingly perfect central caustic region is slower).Thus for small L a cartoon of the evolution is that the initially collapsing ghost ﬁeld causes the higher derivative term to radiate strongly,but the process of radiation simply reﬁnes the collapsing region into the perfect caustic form where it eventually collapses.Suitable modiﬁcation of the conservative equations of motion(4.1)introduce the geometric measure factor associated with axisymmetry.In this case weﬁnd the behaviour essentially analogous to the planar case.Whilst the singular behaviour appears‘weaker’,taking longer to reach the singularity for the same parameters, the singularity does indeed form.However moving to spherical symmetry weﬁnd a change in behaviour.For spherical symmetry we again take the initial data(4.2)and evolve this in the absence of any gravitational potential.Inﬁgure8we plot the evolution of H for π0=0.1and L=0.005(as shown earlier for the planar case).The behaviour is clearly diﬀerent with a totally non-singular evolution for all times,a portion of which is shown in theﬁgure.Whilst for L=0.obviously the spherically symmetric ghost ﬁeld exhibits the usual caustic singularity,we see the higher derivative term,aided by the geometric measure factor,can now radiate suﬃcient energy to avoid any later energy build up.So far we have discussed the case of evolution of a local perturbation in the ghost ﬁeld.We have seen that the symmetry of the initial data has a strong eﬀect on whether the evolution is singular or not.Physically,however,we are interested more。

Evidence of Impending Tipping Point for Earth

Evidence of Impending Tipping Point for EarthJune 6, 2012 — A group of scientists from around the world is warning that population growth, widespread destruction of natural ecosystems, and climate change may be driving Earth toward an irreversible change in the biosphere, a planet-wide tipping point that would have destructive consequences absent adequate preparation and mitigation."It really will be a new world, biologically, at that point," warns Anthony Barnosky, professor of integrative biology at the University of California, Berkeley, and lead author of a review paper appearing in the June 7 issue of the journal Nature. "The data suggests that there will be a reduction in biodiversity and severe impacts on much of what we depend on to sustain our quality of life, including, for example, fisheries, agriculture, forest products and clean water. This could happen within just a few generations."The Nature paper, in which the scientists compare the biological impact of past incidents of global change with processes under way today and assess evidence for what the future holds, appears in an issue devoted to the environment in advance of the June 20-22 United Nations Rio+20 Earth Summit in Rio de Janeiro, Brazil.The result of such a major shift in the biosphere would be mixed, Barnosky noted, with some plant and animal species disappearing, new mixes of remaining species, and major disruptions in terms of which agricultural crops can grow where.The paper by 22 internationally known scientists describes an urgent need for better predictive models that are based on a detailed understanding of how the biosphere reacted in the distant past to rapidly changing conditions, including climate and human population growth. In a related development, ground-breaking research to develop the reliable, detailed biological forecasts the paper is calling for is now underway at UC Berkeley. The endeavor, The Berkeley Initiative in Global Change Biology, or BiGCB, is a massive undertaking involving more than 100 UC Berkeley scientists from an extraordinary range of disciplines that already has received funding: a $2.5 million grant from the Gordon and Betty Moore Foundation and a $1.5 million grant from the Keck Foundation. The paper by Barnosky and others emerged from the first conference convened under the BiGCB's auspices."One key goal of the BiGCB is to understand how plants and animals responded to major shifts in the atmosphere, oceans, and climate in the past, so that scientists can improve their forecasts and policy makers can take the steps necessary to either mitigate or adapt to changes that may be inevitable," Barnosky said. "Better predictive models will lead to better decisions in terms of protecting the natural resources future generations will rely on for quality of life and prosperity."Climate change could also lead to global political instability, according to a U.S. Department of Defense study referred to in the Nature paper."UC Berkeley is uniquely positioned to conduct this sort of complex, multi-disciplinary research," said Graham Fleming, UC Berkeley's vice chancellor for research. "Our world-class museums hold a treasure trove of biological specimens dating back many millennia that tell the story of how our planet has reacted to climate change in the past. That, combined with new technologies and data mining methods used by our distinguished faculty in a broad array of disciplines, will help us decipher the clues to the puzzle of how the biosphere will change as the result of the continued expansion of human activity on our planet."One BiGCB project launched last month, with UC Berkeley scientists drilling into Northern California's Clear Lake, one of the oldest lakes in the world with sediments dating back more than 120,000 years, to determine how past changes in California's climate impacted local plant and animal populations.City of Berkeley Mayor Tom Bates, chair of the Bay Area Joint Policy Committee, said the BiGCB "is providing the type of research that policy makers urgently need as we work to reduce greenhouse gas emissions and prepare the Bay region to adapt to the inevitable impacts of climate change. To take meaningful actions to protect our region, we first need to understand the serious global and local changes that threaten our natural resources and biodiversity.""The Bay Area's natural systems, which we often take for granted, are absolutely critical to the health and well-being of our people, our economy and the Bay Area's quality of life," added Bates. How close is a global tipping point?The authors of the Nature review -- biologists, ecologists, complex-systems theoreticians, geologists and paleontologists from the United States, Canada, South America and Europe -- argue that, although many warning signs are emerging, no one knows how close Earth is to a global tipping point, or if it is inevitable. The scientists urge focused research to identify early warning signs of a global transition and an acceleration of efforts to address the root causes."We really do have to be thinking about these global scale tipping points, because even the parts of Earth we are not messing with directly could be prone to some very major changes," Barnosky said. "And the root cause, ultimately, is human population growth and how many resources each one of us uses."Coauthor Elizabeth Hadly from Stanford University said "we may already be past these tipping points in particular regions of the world. I just returned from a trip to the high Himalayas in Nepal, where I witnessed families fighting each other with machetes for wood -- wood that they wouldburn to cook their food in one evening. In places where governments are lacking basic infrastructure, people fend for themselves, and biodiversity suffers. We desperately need global leadership for planet Earth."The authors note that studies of small-scale ecosystems show that once 50-90 percent of an area has been altered, the entire ecosystem tips irreversibly into a state far different from the original, in terms of the mix of plant and animal species and their interactions. This situation typically is accompanied by species extinctions and a loss of biodiversity.Currently, to support a population of 7 billion people, about 43 percent of Earth's land surface has been converted to agricultural or urban use, with roads cutting through much of the remainder. The population is expected to rise to 9 billion by 2045; at that rate, current trends suggest that half Earth's land surface will be disturbed by 2025. To Barnosky, this is disturbingly close to a global tipping point."Can it really happen? Looking into the past tells us unequivocally that, yes, it can really happen. It has happened. The last glacial/interglacial transition 11,700 years ago was an example of that," he said, noting that animal diversity still has not recovered from extinctions during that time. "I think that if we want to avoid the most unpleasant surprises, we want to stay away from that 50 percent mark."Global change biologyThe paper emerged from a conference held at UC Berkeley in 2010 to discuss the idea of a global tipping point, and how to recognize and avoid it.Following that meeting, 22 of the attendees summarized available evidence of past globalstate-shifts, the current state of threats to the global environment, and what happened after past tipping points.They concluded that there is an urgent need for global cooperation to reduce world population growth and per-capita resource use, replace fossil fuels with sustainable sources, develop more efficient food production and distribution without taking over more land, and better manage the land and ocean areas not already dominated by humans as reservoirs of biodiversity and ecosystem services."Ideally, we want to be able to predict what could be detrimental biological change in time to steer the boat to where we don't get to those points," Barnosky said. "My underlying philosophy is that we want to keep Earth, our life support system, at least as healthy as it is today, in terms of supporting humanity, and forecast when we are going in directions that would reduce our quality of life so that we can avoid that.""My view is that humanity is at a crossroads now, where we have to make an active choice," Barnosky said. "One choice is to acknowledge these issues and potential consequences and try to guide the future (in a way we want to). The other choice is just to throw up our hands and say,'Let's just go on as usual and see what happens.' My guess is, if we take that latter choice, yes, humanity is going to survive, but we are going to see some effects that will seriously degrade the quality of life for our children and grandchildren."The work was supported by UC Berkeley's Office of the Vice Chancellor for Research.。

2025年研究生考试考研英语(一201)试卷及答案指导

2025年研究生考试考研英语(一201)自测试卷及答案指导一、完型填空（10分）Section I: Cloze TestDirections: Read the following text carefully and choose the best answer from the four choices marked A, B, C, and D for each blank.Passage:In today’s rapidly evolving digital landscape, the role of social media has become increasingly significant. Social media platforms are not just tools for personal interaction; they also serve as powerful channels for business promotion and customer engagement. Companies are now leveraging these platforms to reach out to their target audience more effectively than ever before. However, the effectiveness of social media marketing (1)_on how well the company understands its audience and the specific platform being used. For instance, while Facebook may be suitable for reaching older demographics, Instagram is more popular among younger users. Therefore, it is crucial for businesses to tailor their content to fit the preferences and behaviors of the (2)_demographic they wish to target.Moreover, the rise of mobile devices has further transformed the way peopleconsume content online. The majority of social media users now access these platforms via smartphones, which means that companies must ensure that their content is optimized for mobile viewing. In addition, the speed at which information spreads on social media can be both a boon and a bane. On one hand, positive news about a brand can quickly go viral, leading to increased visibility and potentially higher sales. On the other hand, negative publicity can spread just as fast, potentially causing serious damage to a brand’s reputation. As such, it is imperative for companies to have a well-thought-out strategy for managing their online presence and responding to feedback in a timely and professional manner.In conclusion, social media offers unparalleled opportunities for businesses to connect with customers, but it requires careful planning and execution to (3)___the maximum benefits. By staying attuned to trends and continuously adapting their strategies, companies can harness the power of social media to foster growth and build strong relationships with their audiences.1.[A] relies [B] bases [C] stands [D] depends2.[A] particular [B] peculiar [C] special [D] unique3.[A] obtain [B] gain [C] achieve [D] accomplishAnswers:1.D - depends2.A - particular3.C - achieveThis cloze test is designed to assess comprehension and vocabulary skills, as well as the ability to infer the correct usage of words within the context of the passage. Each question is crafted to require understanding of the sentence structure and meaning to select the best option.二、传统阅读理解（本部分有4大题，每大题10分，共40分）第一题Passage:In the 1950s, the United States experienced a significant shift in the way people viewed education. This shift was largely due to the Cold War, which created a demand for a highly educated workforce. As a result, the number of students pursuing higher education in the U.S. began to grow rapidly.One of the most important developments during this period was the creation of the Master’s degree program. The Master’s degree was designed to provide students with advanced knowledge and skills in a specific field. This program became increasingly popular as more and more people realized the value of a higher education.The growth of the Master’s degree program had a profound impact on American society. It helped to create a more educated and skilled workforce, which in turn contributed to the nation’s economic growth. It also helped to improve the quality of life for many Americans by providing them with opportunities for career advancement and personal development.Today, the Master’s degree is still an important part of the American educational system. However, there are some challenges that need to be addressed. One of the biggest challenges is the rising cost of education. As the cost of tuition continues to rise, many students are unable to afford the cost of a Master’s degree. This is a problem that needs to be addressed if we are to continue to provide high-quality education to all Americans.1、What was the main reason for the shift in the way people viewed education in the 1950s?A. The demand for a highly educated workforce due to the Cold War.B. The desire to improve the quality of life for all Americans.C. The increasing cost of education.D. The creation of the Master’s degree program.2、What is the purpose of the Master’s degree program?A. To provide students with basic knowledge and skills in a specific field.B. To provide students with advanced knowledge and skills in a specific field.C. To provide students with job training.D. To provide students with a general education.3、How did the growth of the Master’s degree program impact American society?A. It helped to create a more educated and skilled workforce.B. It helped to improve the quality of life for many Americans.C. It caused the economy to decline.D. It increased the cost of education.4、What is one of the biggest challenges facing the Master’s deg ree program today?A. The demand for a highly educated workforce.B. The rising cost of education.C. The desire to improve the quality of life for all Americans.D. The creation of new educational programs.5、What is the author’s main point in the last pa ragraph?A. The Master’s degree program is still an important part of the American educational system.B. The cost of education needs to be addressed.C. The Master’s degree program is no longer relevant.D. The author is unsure about the future of the Master’s degree program.第二题Reading Comprehension (Traditional)Passage:The digital revolution has transformed the way we live, work, and communicate. With the advent of the internet and the proliferation of smart devices, information is more accessible than ever before. This transformation has had a profound impact on education, with online learning platforms providing unprecedented access to knowledge. However, this shift towards digital learningalso poses challenges, particularly in terms of ensuring equitable access and maintaining educational quality.While the benefits of digital learning are numerous, including flexibility, cost-effectiveness, and the ability to reach a wider audience, there are concerns about the potential for increased social isolation and the difficulty in replicating the dynamic, interactive environment of a traditional classroom. Moreover, not all students have equal access to the technology required for online learning, which can exacerbate existing inequalities. It’s crucial that as we embrace the opportunities presented by digital technologies, we also address these challenges to ensure that no student is left behind.Educators must adapt their teaching methods to take advantage of new tools while also being mindful of the need to foster a sense of community and support among students. By integrating both digital and traditional approaches, it’s possible to create a learning environment that leverages the strengths of each, ultimately enhancing the educational experience for all students.Questions:1、What is one of the main impacts of the digital revolution mentioned in the passage?•A) The reduction of social interactions•B) The increase in physical book sales•C) The transformation of communication methods•D) The decline of online learning platformsAnswer: C) The transformation of communication methods2、According to the passage, what is a challenge associated with digital learning?•A) The inability to provide any form of interaction•B) The potential to widen the gap between different socioeconomic groups •C) The lack of available content for online courses•D) The complete replacement of traditional classroomsAnswer: B) The potential to widen the gap between different socioeconomic groups3、Which of the following is NOT listed as a benefit of digital learning in the passage?•A) Cost-effectiveness•B) Flexibility•C) Increased social isolation•D) Wider reachAnswer: C) Increased social isolation4、The passage suggests that educators should do which of the following in response to the digital revolution?•A) Abandon all traditional teaching methods•B) Focus solely on improving students’ technical skills•C) Integrate digital and traditional teaching methods•D) Avoid using any digital tools in the classroomAnswer: C) Integrate digital and traditional teaching methods5、What is the author’s stance on the role of digital technologies ineducation?•A) They are unnecessary and should be avoided•B) They offer opportunities that should be embraced, but with caution •C) They are the only solution to current educational challenges•D) They have no real impact on the quality of educationAnswer: B) They offer opportunities that should be embraced, but with cautionThis reading comprehension exercise is designed to test your understanding of the text and your ability to identify key points and arguments within the passage.第三题Reading PassageWhen the French sociologist and philosopher Henri Lefebvre died in 1991, he left behind a body of work that has had a profound influence on the fields of sociology, philosophy, and cultural studies. Lefebvre’s theories focused on the relationship between space and society, particularly how space is produced, represented, and experienced. His work has been widely discussed and debated, with scholars and critics alike finding value in his insights.Lefebvre’s most famous work, “The Production of Space,” published in 1974, laid the foundation for his theoretical framework. In this book, he argues that space is not simply a container for human activities but rather an active agent in shaping social relationships and structures. Lefebvre introduces the concept of “three spaces” to describe the production of space: the perceived space,the lived space, and the representative space.1、According to Lefebvre, what is the primary focus of his theories?A. The development of urban planningB. The relationship between space and societyC. The history of architectural designD. The evolution of cultural practices2、What is the main argument presented in “The Production of Space”?A. Space is a passive entity that reflects social structures.B. Space is a fundamental building block of society.C. Space is an object that can be easily manipulated by humans.D. Space is irrelevant to the functioning of society.3、Lefebvre identifies three distinct spaces. Which of the following is NOT one of these spaces?A. Perceived spaceB. Lived spaceC. Representative spaceD. Economic space4、How does Lefebvre define the concept of “three spaces”?A. They are different types of architectural designs.B. They represent different stages of the production of space.C. They are different ways of perceiving and experiencing space.D. They are different social classes that occupy space.5、What is the significance of Lefebvre’s work in the fields of sociology and philosophy?A. It provides a new perspective on the role of space in social relationships.B. It offers a comprehensive guide to urban planning and development.C. It promotes the idea that space is an unimportant aspect of society.D. It focuses solely on the history of architectural movements.Answers:1、B2、B3、D4、C5、A第四题Reading Comprehension (Traditional)Read the following passage and answer the questions that follow. Choose the best answer from the options provided.Passage:In recent years, there has been a growing interest in the concept of “smart cities,” which are urban areas that u se different types of electronic data collection sensors to supply information which is used to manage assets and resources efficiently. This includes data collected from citizens, devices, andassets that is processed and analyzed to monitor and manage traffic and transportation systems, power plants, water supply networks, waste management, law enforcement, information systems, schools, libraries, hospitals, and other community services. The goal of building a smart city is to improve quality of life by using technology to enhance the performance and interactivity of urban services, to reduce costs and resource consumption, and to increase contact between citizens and government. Smart city applications are developed to address urban challenges such as environmental sustainability, mobility, and economic development.Critics argue, however, that while the idea of a smart city is appealing, it raises significant concerns about privacy and security. As more and more aspects of daily life become digitized, the amount of personal data being collected also increases, leading to potential misuse or unauthorized access. Moreover, the reliance on technology for critical infrastructure can create vulnerabilities if not properly secured against cyber-attacks. There is also a risk of widening the digital divide, as those without access to the necessary technologies may be left behind, further exacerbating social inequalities.Despite these concerns, many governments around the world are moving forward with plans to develop smart cities, seeing them as a key component of their future strategies. They believe that the benefits of improved efficiency and service delivery will outweigh the potential risks, provided that adequate safeguards are put in place to protect citizen s’ data and ensure the resilience of thecity’s technological framework.Questions:1、What is the primary purpose of developing a smart city?•A) To collect as much data as possible•B) To improve the quality of life through efficient use of technology •C) To replace all traditional forms of communication•D) To eliminate the need for human interaction in urban services2、According to the passage, what is one of the main concerns raised by critics regarding smart cities?•A) The lack of available technology•B) The high cost of implementing smart city solutions•C) Privacy and security issues related to data collection•D) The inability to provide essential services3、Which of the following is NOT mentioned as an area where smart city technology could be applied?•A) Traffic and transportation systems•B) Waste management•C) Educational institutions•D) Agricultural production4、How do some governments view the development of smart cities despite the criticisms?•A) As a risky endeavor that should be avoided•B) As a temporary trend that will soon pass•C) As a strategic move with long-term benefits•D) As an unnecessary investment in technology5、What does the term “digital divide” refer to in the context of smart cities?•A) The gap between the amount of data collected and the amount of data analyzed•B) The difference in technological advancement between urban and rural areas•C) The disparity in access to technology and its impact on social inequality•D) The separation of digital and non-digital methods of service delivery Answers:1、B) To improve the quality of life through efficient use of technology2、C) Privacy and security issues related to data collection3、D) Agricultural production4、C) As a strategic move with long-term benefits5、C) The disparity in access to technology and its impact on social inequality三、阅读理解新题型（10分）Reading Comprehension (New Type)Passage:The rise of e-commerce has transformed the way people shop and has had aprofound impact on traditional brick-and-mortar retailers. Online shopping offers convenience, a wide range of products, and competitive prices. However, it has also raised concerns about the future of physical stores. This passage examines the challenges and opportunities facing traditional retailers in the age of e-commerce.In recent years, the popularity of e-commerce has soared, thanks to advancements in technology and changing consumer behavior. According to a report by Statista, global e-commerce sales reached nearly$4.2 trillion in 2020. This upward trend is expected to continue, with projections showing that online sales will account for 25% of total retail sales by 2025. As a result, traditional retailers are facing fierce competition and must adapt to the digital landscape.One of the main challenges for brick-and-mortar retailers is the shift in consumer preferences. Many shoppers now prefer the convenience of online shopping, which allows them to compare prices, read reviews, and purchase products from the comfort of their homes. This has led to a decrease in foot traffic in physical stores, causing many retailers to struggle to attract customers. Additionally, the ability to offer a wide range of products at competitive prices has become a hallmark of e-commerce, making it difficult for traditional retailers to compete.Despite these challenges, there are opportunities for traditional retailers to thrive in the age of e-commerce. One approach is to leverage the unique strengths of physical stores, such as the ability to provide an immersiveshopping experience and personalized customer service. Retailers can also use technology to enhance the in-store experience, such as implementing augmented reality (AR) to allow customers to visualize products in their own homes before purchasing.Another strategy is to embrace the digital world and create a seamless shopping experience that integrates online and offline channels. For example, retailers can offer online returns to brick-and-mortar stores, allowing customers to shop online and return items in person. This not only provides convenience but also encourages customers to make additional purchases while they are in the store.Furthermore, traditional retailers can leverage their established brand loyalty and customer base to create a competitive advantage. By focusing on niche markets and offering unique products or services, retailers can differentiate themselves from e-commerce giants. Additionally, retailers can invest in marketing and promotions to drive traffic to their physical stores, even as more consumers turn to online shopping.In conclusion, the rise of e-commerce has presented traditional retailers with significant challenges. However, by embracing the digital landscape, leveraging their unique strengths, and focusing on customer satisfaction, traditional retailers can adapt and thrive in the age of e-commerce.Questions:1.What is the main concern raised about traditional retailers in the age of e-commerce?2.According to the passage, what is one of the main reasons for the decline in foot traffic in physical stores?3.How can traditional retailers leverage technology to enhance the in-store experience?4.What strategy is mentioned in the passage that involves integrating online and offline channels?5.How can traditional retailers create a competitive advantage in the age of e-commerce?Answers:1.The main concern is the fierce competition from e-commerce and the shift in consumer preferences towards online shopping.2.The main reason is the convenience and competitive prices offered by e-commerce, which make it difficult for traditional retailers to compete.3.Traditional retailers can leverage technology by implementing augmented reality (AR) and offering online returns to brick-and-mortar stores.4.The strategy mentioned is to create a seamless shopping experience that integrates online and offline channels, such as offering online returns to brick-and-mortar stores.5.Traditional retailers can create a competitive advantage by focusing on niche markets, offering unique products or services, and investing in marketing and promotions to drive traffic to their physical stores.四、翻译（本大题有5小题，每小题2分，共10分）First QuestionTranslate the following sentence into Chinese. Write your translation on the ANSWER SHEET.Original Sentence:“Although technology has brought about nume rous conveniences in our daily lives, it is also true that it has led to significant privacy concerns, especially with the rapid development of digital communication tools.”Answer:尽管技术在我们的日常生活中带来了诸多便利，但也不可否认它导致了重大的隐私问题，尤其是在数字通信工具快速发展的情况下。

杨振宁诺贝尔奖演讲词

C H E N N I N G Y A N GThe law of parity conservation and othersymmetry laws of physicsNobel Lecture, December 11, 1957It is a pleasure and a great privilege to have this opportunity to discuss with you the question of parity conservation and other symmetry laws. We shall be concerned first with the general aspects of the role of the symmetry laws in physics; second, with the development that led to the disproof of parity conservation; and last, with a discussion of some other symmetry laws which physicists have learned through experience, but which do not yet together form an integral and conceptually simple pattern. The interesting and very exciting developments since parity conservation was disproved, will be cov-ered by Dr. Lee in his lecture1.IThe existence of symmetry laws is in full accordance with our daily ex-perience. The simplest of these symmetries, the isotropy and homogeneity of space, are concepts that date back to the early history of human thought. The invariance of physical laws under a coordinate transformation of uni-form velocity, also known as the invariance under Galilean transformations, is a more sophisticated symmetry that was early recognized, and formed one of the corner-stones of Newtonian mechanics. Consequences of these sym-metry principles were greatly exploited by physicists of the past centuries and gave rise to many important results. A good example in this direction is the theorem that in an isotropic solid there are only two elastic constants. Another type of consequences of the symmetry laws relates to the con-servation laws. It is common knowledge today that in general a symmetry principle (or equivalently an invariance principle) generates a conservation law. For example, the invariance of physical laws under space displacement has as a consequence the conservation of momentum, the invariance under space rotation has as a consequence the conservation of angular momentum. While the importance of these conservation laws was fully understood, their close relationship with the symmetry laws seemed not to have been clearly recognized until the beginning of the twentieth century2. (Cf. Fig. 1.)3941957C.N.Y A N GFig. 1 .With the advent of special and general relativity, the symmetry laws gained new importance. Their connection with the dynamic laws of physics takes on a much more integrated and interdependent relationship than in classical mechanics, where logically the symmetry laws were only conse-quences of the dynamical laws that by chance possess the symmetries. Also in the relativity theories the realm of the symmetry laws was greatly en-riched to include invariances that were by no means apparent from daily experience. Their validity rather was deduced from, or was later confirmed by complicated experimentation. Let me emphasize that the conceptual sim-plicity and intrinsic beauty of the symmetries that so evolve from complex experiments are for the physicists great sources of encouragement. One learns to hope that Nature possesses an order that one may aspire to comprehend. It was, however, not until the development of quantum mechanics that the use of the symmetry principles began to permeate into the very language of physics. The quantum numbers that designate the states of a system are often identical with those that represent the symmetries of the system. It in-deed is scarcely possible to overemphasize the role played by the symmetry principles in quantum mechanics. To quote two examples: The general struc-ture of the Periodic Table is essentially a direct consequence of the isotropy of Coulomb’s law. The existence of the antiparticles - namely the positron, the antiproton, and the antineutron - were theoretically anticipated as con-sequences of the symmetry of physical laws with respect to Lorentz trans-formations. In both cases Nature seems to take advantage of the simple mathematical representations of the symmetry laws. When one pauses to consider the elegance and the beautiful perfection of the mathematical rea-soning involved and contrast it with the complex and far-reaching physicalP A R I T Y C O N S E R V A T I O N A N D O T H E R S Y M M E T R Y L A W S395 consequences, a deep sense of respect for the power of the symmetry laws never fails to develop.One of the symmetry principles, the symmetry between the left and the right, is as old as human civilization. The question whether Nature exhibits such symmetry was debated at length by philosophers of the pasts. Of course, in daily life, left and right are quite distinct from each other. Our hearts, for example, are on our left sides. The language that people use both in the orient and the occident, carries even a connotation that right is good and left is evil. However, the laws of physics have always shown complete symmetry between the left and the right, the asymmetry in daily life being attributed to the accidental asymmetry of the environment, or initial conditions in organic life. To illustrate the point, we mention that if there existed a mirror-image man with his heart on his right side, his internal organs reversed com-pared to ours, and in fact his body molecules, for example sugar molecules, the mirror image of ours, and if he ate the mirror image of the food that we eat, then according to the laws of physics, he should function as well as we do. The law of right-left symmetry was used in classical physics, but was not of any great practical importance there. One reason for this derives from the fact that right-left symmetry is a discrete symmetry, unlike rotational sym-metry which is continuous. Whereas the continuous symmetries always lead to conservation laws in classical mechanics, a discrete symmetry does not. With the introduction of quantum mechanics, however, this difference between the discrete and continuous symmetries disappears. The law of right-left symmetry then leads also to a conservation law: the conservation of parity. The discovery of this conservation law dates back to 1924 when Laporte4 found that energy levels in complex atoms can be classified into « gestriche-ne » and « ungestrichene » types, or in more recent language, even and odd levels. In transitions between these levels during which one photon is emitted or absorbed, Laporte found that the level always changes from even to odd or vice versa. Anticipating later developments, we remark that the evenness or oddness of the levels was later referred to as the parity of the levels. Even levels are defined to have parity +1,odd levels parity -1. One also defines the photon emitted or absorbed in the usual atomic transitions to have odd parity. Laporte’s rule can then be formulated as the statement that in an atomic transition with the emission of a photon, the parity of the initial state is equal to the total parity of the final state, i.e. the product of the parities of the final atomic state and the photon emitted. In other words, parity is conserved, or unchanged, in the transition.3961957 C. N. YANGIn 1927 Wigners took the critical and profound step to prove that the empirical rule of Laporte is a consequence of the reflection invariance, or right-left symmetry, of the electromagnetic forces in the atom. This fun-damental idea was rapidly absorbed into the language of physics. Since right-left symmetry was unquestioned also in other interactions, the idea was fur-ther taken over into new domains as the subject matter of physics extended into nuclear reactions,puzzle developed in the last few years. Before explaining the meaning of this puzzle it is best to go a little bit into a classification of the forces that act between subatomic particles, a classification which the physicists have learned through experience to use in the last 50 years. We list the four classes of interactions below. The strength of these interactions is indicated in the column on the right.The strongest interactions are the nuclear interactions which include the forces that bind nuclei together and the interaction between the nuclei and theP A R I T Y C O N S E R V A T I O N A N D O T H E R S Y M M E T R Y L A W S397 this century in the β-radioactivity of nuclei, a phenomena which especially in the last 25 years has been extensively studied experimentally. With the discovery of decays and µ capture it was noticed independently6 by Klein, by Tiomno and Wheeler, and by Lee, Rosenbluth and me, that these interactions have roughly the same strengths as β-interactions. They are called weak interactions, and in the last few years their rank has been con-stantly added to through the discovery of many other weak interactions responsible for the decay of the strange particles. The consistent and striking pattern of their almost uniform strength remains today one of the most tan-talizing phenomena - a topic which we shall come back to later. About the last class of forces, the gravitational forces, we need only mention that in atomic and nuclear interactions they are so weak as to be completely neg-ligible in all the observations with existing techniques.Now to return to theand τ mesonssome information about the spins and parities of the τ andmeson must have the total parity, or in other words, the product parity, of two π mesons,which is even (i.e. = +1). Similarly, the τ meson must have the total parity of three π mesons, which is odd. Actually because of the relative motion of the π mesons the argument was not as simple and unambiguous as we just discussed. To render the ar-gument conclusive and definitive it was necessary to study experimentally the momentum and angular distribution of the π mesons. Such studies were made in many laboratories, and by the spring of 1956 the accumulated ex-perimental data seemed to unambiguously indicate, along the lines of rea-soning discussed above, thatϑ and τ do not have the same parity, and con-sequently are not the same particle. This conclusion, however, was in marked contradiction with other experimental results which also became definite at about the same time. The contradiction was known as the ϑ-τ puzzle and was widely discussed. To recapture the atmosphere of that time allow me to quote a paragraph concerning the conclusion that3981957C.N.Y A N Gparticle from a report entitled « Present Knowledge about the New Par-ticles » which I gave at the International Conference on Theoretical Physics8 in Seattle, in September 1956.« However it will not do to jump to hasty conclusions. This is because ex-perimentally the K mesons (i.e. τ and ϑ) seem all to have the same masses and the same lifetimes. The masses are known to an accuracy of, say, from 2 to 10electron masses, or a fraction of a percent, and the lifetimes are known to an accuracy of, say, 20 percent. Since particles which have different spin and parity values, and which have strong interactions with the nucleons and pions, are not expected to have identical masses and lifetimes, one is forced to keep the question open whether the inference mentioned above that the are not the same particle is conclusive. Parenthetically, I might addthat the inference would certainly have been regarded as conclusive, and in fact more well-founded than many inferences in physics, had it not been for the anomaly of mass and lifetime degeneracies. »The situation that the physicist found himself in at that time has been likened to a man in a dark room groping for an outlet. He is aware of the fact that in some direction there must be a door which would lead him out of his predicament. But in which direction?That direction turned out to lie in the faultiness of the law of parity con-servation for the weak interactions. But to uproot an accepted concept one must first demonstrate why the previous evidence in its favor were insuffi-cient. Dr. Lee and I9 examined this question in detail, and in May 1956 we came to the following conclusions: (A) Past experiments on the weak inter-actions had actually no bearing on the question of parity conservation. (B) In the strong interactions, i.e. interactions of classes 1and 2 discussed above, there were indeed many experiments that established parity conservation to a high degree of accuracy, but not to a sufficiently high degree to be able to reveal the effects of a lack of parity conservation in the weak interactions. The fact that parity conservation in the weak interactions was believed for so long without experimental support was very startling. But what was more startling was the prospect that a space-time symmetry law which the phys-icists have learned so well may be violated. This prospect did not appeal to us. Rather we were, so to speak, driven to it through frustration with the various other efforts at understanding theP A R I T Y C O N S E R V A T I O N A N D O T H E R S Y M M E T R Y L A W S399 an approximate symmetry law was, however, not expected of the sym-metries related to space and time. In fact one is tempted to speculate, now that parity conservation is found to be violated in the weak interactions, whether in the description of such phenomena the usual concept of space and time is adequate. At the end of our discussion we shall have the occasion to come back to a closely related topic.Why was it so that among the multitude of experiments onThis experiment was first performed in the latter half of 1956 and finished early this year by Wu, Ambler, Hayward, Hoppes, and Hudson12. The actual experimental setup was very involved, because to eliminate disturbing out-side influences the experiment had to be done at very low temperatures. The technique of combining β-decay measurement with low temperature ap-paratus was unknown before and constituted a major difficulty which was successfully solved by these authors. To their courage and their skill, phys-icists owe the exciting and clarifying developments concerning parity con-servation in the past year.of cobalt. Very rapidly after these results were made known, many experi-ments were performed which further demonstrated the violation of parity conservation in various weak interactions. In his lecturer Dr. Lee will discuss these interesting and important developments.I I IThe breakdown of parity conservation brings into focus a number of ques-tions concerning symmetry laws in physics which we shall now briefly dis-cuss in general terms:(A) As Dr. Lee1 will discuss, the experiment of Wu, Ambler, and their collaborators also proves13,14 that charge conjugation invariance15 is violated forP A R I T Y C O N S E R V A T I O N A N D O T H E R S Y M M E T R Y L A W S401 The three discrete invariances - reflection invariance, charge conjugation invariance, and time reversal invariance - are connected by an important theorem17 called the CPT theorem. Through the use of this theorem one can prove13 a number of general results concerning the experimental manifesta-tions of the possible violations of the three symmetries in the weak inter-actions.Of particular interest is the possibility that time reversal invariance in the weak interactions may turn out to be intact. If this is the case, it follows from the CPT theorem that although parity conservation breaks down, right-left symmetry will still hold if18 one switches all particles into antiparticles in taking a mirror image.In terms of Fig. 2 this means that if one changes all the matter that composes the apparatus at the right into anti-matter, the meter reading would become the same for the two sides if time reversal invariance holds. It is important to notice that in the usual definition of re-flection, the electric field is a vector and the magnetic field a pseudovector while in this changed definition their transformation properties are switched. The transformation properties of the electric charge and the magnetic charge are also interchanged. It would be interesting to speculate on the possible relationship between the nonconservation of parity and the symmetrical or unsymmetrical role played by the electric and magnetic fields.The question of the validity of the continuous space time symmetry laws has been discussed to some extent in the past year. There is good evidence that these symmetry laws do not break down in the weak interactions. (B) Another symmetry law that has been widely discussed is that giving rise to the conservation of isotopic spin19. In recent years the use of this sym-metry law has produced a remarkable empirical order among the phenom-ena concerning the strange particles20.It is however certainly the least under-stood of all the symmetry laws. Unlike Lorentz invariance or reflection invariance, it is not a « geometrical » symmetry law relating to space time invariance properties. Unlike charge conjugation invariance21 it does not seem to originate from the algebraic property of the complex numbers that occurs in quantum mechanics. In these respects it resembles the conservation laws of charge and heavy particles. These latter laws, however, are exact while the conservation of isotopic spin is violated upon the introduction of electromagnetic interactions and weak interactions. An understanding of the origin of the conservation of isotopic spin and how to integrate it with the other symmetry laws is undoubtedly one of the outstanding problems in high-energy physics today.4021957 C.N.Y A N G(C) We have mentioned before that all the different varieties of weak interactions share the property of having very closely identical strengths. The experimental work on parity nonconservation in the past year reveals that they very likely also share the property of not respecting parity conservation and charge conjugation invariance. They therefore serve to differentiate be-tween right and left once one fixes one’s definition of matter vs. anti-mat-ter. One could also use the weak interactions to differentiate between matter and anti-matter once one chooses a definition of right vs. left. If time rever-sal invariance is violated, the weak interactions may even serve to differen-tiate simultaneously right from left, and matter from anti-matter. One senses herein that maybe the origin of the weak interactions is intimately tied in with the question of the differentiability of left from right, and of matter from anti-matter.1. T. D. Lee, Nobel Lecture, this volume, p. 406.2.For references to these developments see E. P. Wigner, Proc. Am. Phil. Soc., 93(1949) 521.3. Cf. the interesting discussion on bilateral symmetry by H. Weyl, Symmetry, Prince-ton University Press, 1952.4. O. Laporte, Z.Physik, 23 (1924) 135.5. E. P. Wigner, Z. Physik, 43 (1927) 624.6. O. Klein, Nature, 161 (1948) 897; J. Tiomno and J. A. Wheeler, Rev.Mod. Phys.,21 (1949) 144;T. D. Lee, M. Rosenbluth, and C. N. Yang, Phys. Rev., 75 (1949)905.7. R. Dalitz, Phil. Mag., 44 (1953) 1068; E. Fabri, Nuovo Cimento, II(1954) 479.8. C. N. Yang, Rev. Mod. Phys., 29 (1957) 231.9. T. D. Lee and C. N. Yang, Phys. Rev., 104 (1956) 254.10. T. D. Lee and J. Orear, Phys. Rev., 100 (1955) 932;T. D. Lee and C. N. Yang,Phys. Rev., 102 (1956) 290; M. Gell-Mann, (unpublished); R. Weinstein, (private communication) ; a general discussion of these ideas can be found in the Proceedings of the Rochester Conference, April 1956, Session VIII, Interscience, New York, 1957.11. C. N. Yang and J. Tiomno, Phys. Rev., 79 (1950) 495.12. C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson, Phys.Rev.,105 (1957) 1413.13. T. D. Lee, R. Oehme, and C. N. Yang, Phys. Rev., 106 (1957) 340.14. B. L. Ioffe, L. B. Okun, and A. P. Rudik, J.E.T.P. (U.S.S.R.), 32 (1957) 396.English translation in Soviet Phys. ]ETP, 5 (1957) 328.15. Charge conjugation invariance is very intimately tied with the hole theory inter-pretation of Dirac’s equation. The development of the latter originated with P. A.M. Dirac, Proc. Roy. Soc. London, A126 (1930) 360; J. R. Oppenheimer, Phys. Rev.,P A R I T Y C O N S E R V A T I O N A N D O T H E R S Y M M E T R Y L A W S40335 (1930) 562 and H. Weyl, Gruppentheorie und Quantenmechanik, 2nd ed., 1931,p. 234. An account of these developments is found in P. A. M. Dirac, Proc. Roy.S O c. London, A133(1931) 60. Detailed formalism and application of charge con-jugation invariance started with H. A. Kramers, Proc. Acad. Sci. Amsterdam, 40 (1937) 814and W. Furry, Phys. Rev., 51 (1937) 125.16.E. P. Wigner, Nachr. Akad. Wiss. Goettingen, Math.-Physik., 1932, p. 546.Thispaper explains in terms of time reversal invariance the earlier work of H. Kramers, Proc. Acad. Sci. Amsterdam, 33 (1930) 959.17.J. Schwinger, Phys. Rev., 91 (1953) 720, 723;G. Lüders, Kgl. Danske Videnskab.au‘s article in Niels Bohr and the Selskab., Mat.-Fys. Medd., 28, No. 5 (1954);W. P liDevelopment of Physics, Pergamon Press, London, 1955. See also Ref. 21.18.This possibility was discussed by T. D. Lee and C. N. Yang and reported by C. N.Yang at the International Conference on Theoretical Physics in Seattle in Septem-ber 1956. (See Ref. 8.) Its relation with the CPT theorem was also reported in the same conference in one of the discussion sessions. The speculation was later pub-lished in T. D. Lee and C. N. Yang, Phys. Rev., 105(1957) 1671. Independently the possibility has been advanced as the correct one by L. Landau, J.E.T.P.(U.S.S.R.), 32 (1957) 405. An English translation of Landau’s article appeared in Soviet Phys. JETP, 5 (1957) 336.19. The concept of a total isotopic spin quantum number was first discussed by B.Cassen and E. U. Condon, Phys. Rev., 50(1936) 846and E. P. Wigner, Phys. Rev., 51(1937) 106.The physical basis derived from the equivalence of p-p and n-p forces, pointed out by G. Breit, E. U. Condon, and R. D. Present, Phys. Rev., 50 (1936) 825. The isotopic spin was introduced earlier as a formal mathematical parameter by W. Heisenberg, Z. Physik, 77 (1932) I.20.A. Pais, Phys. Rev., 86 (1952) 663, introduced the idea of associated production ofstrange particles. An explanation of this phenomenon in terms of isotopic spin conservation was pointed out by M. Gell-Mann, Phys. Rev., 92 (1953) 833and by K. Nishijima, Progr. Theoret. Phys. (Kyoto), 12 (1954) 107.These latter authors also showed that isotopic spin conservation leads to a convenient quantum number called strangeness.21.R. Jost, Helv. Phys. Acta, 30 (1957) 409.。

高二英语科学概念高级应用单选题45题

高二英语科学概念高级应用单选题45题1.The force acting on an object is directly proportional to its acceleration. If the force on an object is doubled, its acceleration will _____.A.be halvedB.remain the sameC.be doubledD.be quadrupled答案：C。

解析：根据牛顿第二定律F=ma，力与加速度成正比。

当力加倍时，加速度也会加倍。

A 选项加速度减半错误；B 选项加速度不变错误；D 选项加速度变为四倍错误。

2.In an electromagnetic field, the direction of the magnetic field is perpendicular to the direction of the electric field. If the electric field points north, the magnetic field cannot point _____.A.eastB.southC.westD.north答案：D。

解析：因为磁场方向与电场方向垂直，所以当电场指向北时，磁场不能指向北。

A 选项东与北垂直，有可能；B 选项南与北垂直，有可能；C 选项西与北垂直，有可能。

3.When an object is in equilibrium, the sum of all forces acting on it is _____.A.zeroB.greater than zeroC.less than zeroD.indeterminate答案：A。

解析：物体处于平衡状态时，所受合力为零。

B 选项大于零错误，物体将有加速度；C 选项小于零错误，同理；D 选项不确定错误。

4.The work done on an object is equal to the force applied multiplied by the distance moved in the direction of the force. If a force of 10 N moves an object 5 meters in its direction, the work done is _____.A.2 JB.50 JC.15 JD.25 J答案：B。

圆锥曲线论英文

圆锥曲线论英文Conic sections are a fundamental topic in mathematics that deals with the properties and equations of curves formed by the intersection of a plane with a cone. These curves include the circle, ellipse, parabola, and hyperbola. In this document, we will explore the characteristics and equations of these conic sections.1. Circle:A circle is a conic section formed when a plane intersects a cone at a right angle to its axis. It is defined as the set of all points in a plane that are equidistant from a fixed center point. The equation of a circle with center (h, k) and radius r is given by (x h)^2 + (y k)^2 = r^2.2. Ellipse:An ellipse is formed when a plane intersects a cone at an angle that is less than a right angle. It is defined as the set of all points in a plane for which the sum of the distances from two fixed points (called foci) is constant. The equation of an ellipse with center (h, k), major axis length 2a, and minor axis length 2b is given by ((x h)^2 / a^2) + ((y k)^2 / b^2) = 1.3. Parabola:A parabola is formed when a plane intersects a cone parallel to one of its generating lines. It is defined as the set of all points in a plane that are equidistant from a fixed point (called the focus) and a fixed line (called the directrix). The equation of a parabola with vertex (h, k) and focal length p is given by (x h)^2 = 4p(y k).4. Hyperbola:A hyperbola is formed when a plane intersects a cone at an angle greater than a right angle. It is defined as the set of all points in a plane for which the absolute value of the difference of the distances from two fixed points (called foci) is constant. The equation ofa hyperbola with center (h, k), transverse axis length 2a, and conjugate axis length 2b is given by ((x h)^2 / a^2) ((y k)^2 / b^2) = 1.These equations provide a mathematical representation of the conic sections and allow us to analyze their properties. By manipulating these equations, we can determine important characteristics such as the shape, size, orientation, and position of the conic sections.In addition to their geometric properties, conic sections have various applications in different fields. For example, circles are commonly used in geometry, physics, and engineering to represent objects with rotational symmetry. Ellipses are used in astronomy to describe the orbits of planets and satellites. Parabolas are used in physics to model the trajectory of projectiles and in engineering to design reflectors and antennas. Hyperbolas are used in physics and engineering to describe the behavior of waves and particles.In conclusion, conic sections are a fascinating topic in mathematics with diverse applications in various fields. Understanding the properties and equations of circles, ellipses, parabolas, and hyperbolas allows us to analyze and solve problems involving these curves. By studying conic sections, we gain valuable insights into the fundamental principles of geometry and their practical applications.。

A First Course in Probability

A First Course in ProbabilityIntroductionProbability theory is a branch of mathematics that deals with the study of randomness and uncertainty. It provides a framework for understanding and quantifying uncertainties in various fields, ranging from finance and economics to engineering and science. A First Course in Probability aims to introduce the fundamental concepts and techniques of probability theory and provide a solid foundation for further study in the subject.Basic Probability TheorySample Space and EventsIn probability theory, we start by defining a sample space, denoted by Ω, which is the set of all possible outcomes of an experiment. An event, denoted by A, is a subset of the sample space. The probability of an event is a real number between 0 and 1, representing the likelihood of that event occurring.The Calculus of ProbabilityThe basic operations of probability include union, intersection, and complement. Given two events A and B, the union of A and B, denoted by A ∪ B, consists of a ll outcomes that belong to either A or B. The intersection of A and B, denoted by A ∩ B, consists of all outcomes that belong to bothA and B. The complement of an event A, denoted by A’, consists of all outcomes that do not belong to A.The probability of the union of two events is given by the sum of their individual probabilities minus the probability of their intersection:P(A ∪ B) = P(A) + P(B) - P(A ∩ B)Conditional ProbabilityConditional probability measures the likelihood of an event A occurring given that another event B has already occurred. It is denoted by P(A|B) and is defined as:P(A|B) = P(A ∩ B) / P(B), where P(B) > 0IndependenceTwo events A and B are said to be independent if the occurrence of one event does not affect the probability of the other event. Mathematically, two events are independent if and only if:P(A ∩ B) = P(A) * P(B)Random VariablesA random variable is a function that assigns a real number to each outcome in the sample space. It provides a way to quantify the uncertainty associated with an experiment. Random variables can be discrete or continuous, depending onwhether they take on a countable or uncountable number of values, respectively.Probability DistributionsDiscrete Probability DistributionsIn the case of discrete random variables, the probability distribution can be defined by a probability mass function (PMF), which gives the probability of each possible value of the random variable. The PMF satisfies two properties: it must be non-negative for all values of the random variable, and the sum of the probabilities must equal 1.Examples of discrete probability distributions include the Bernoulli distribution, the binomial distribution, and the Poisson distribution.Continuous Probability DistributionsFor continuous random variables, the probability distribution is defined by a probability density function (PDF), which specifies the relative likelihood of the random variable taking on different values. The PDF must be non-negative, and the total area under the curve must equal 1.Examples of continuous probability distributions include the normal distribution, the exponential distribution, and the uniform distribution.Expectation and VarianceExpectationThe expectation of a random variable, denoted by E(X), is a measure of its average value. For discrete random variables, the expectation is calculated by summing the product of each possible value and its corresponding probability. For continuous random variables, the expectation is calculated by integrating the product of each value and its corresponding density over the entire range of values.VarianceThe variance of a random variable, denoted by Var(X), measures the spread or dispersion of its probability distribution. It quantifies how far the values of the random variable deviate from its expectation. The variance is calculated by taking the expectation of the squared difference between each value and the expectation.Central Limit TheoremThe central limit theorem states that the sum or average of a large number of independent and identically distributed random variables will be approximately normally distributed, regardless of the shape of the original distribution. This theorem has wide-ranging applications in statistics and allows us to make inferences about population parameters based on sample data.ConclusionA First Course in Probability provides a solid foundation in the fundamental concepts and techniques of probability theory. It covers basic probability theory, probability distributions, expectation and variance, and the central limit theorem. This course serves as a starting point for further study in the field of probability and its applications.。

完整性与功利主义伦理学

完整性与功利主义伦理学关于《完整性与功利主义伦理学》，是我们特意为大家整理的，希望对大家有所帮助。

摘要：伯纳德威廉斯强调哲学必须直面人类生活的复杂性和困难性，他批判以往哲学回避现实，特别是功利主义和康德主义，以恶劣的方式将生活简单化，忽视了个人情感、规划和运气对伦理生活的影响。

他通过对功利主义后果论结构的分析，指出其中所包含的消极责任特征，批判功利主义忽视了个人分离性的重要性以及对个人完整性的破坏。

威廉斯强调个人是情感需要、功利偏好和理性能力的综合体，主张应该从人的主体自身出发去思考道德问题，认为重要问题是人如何过有意义的生活，而不是我应该遵守什么样的规则。

威廉斯的批判是强有力的，对功利主义的发展具有重要启发。

下载论文网关键词：伯纳德威廉斯；个人完整性；后果论；消极责任；不偏不倚；功利主义中图分类号：B561.6 文献标识码：A 文章编号：1672-3104(2014)03?A50?A8一、关于后果论的理论架构及其缺陷经过古典功利主义与现代功利主义的发展演进，功利主义的理论形态发生了很大的变化，但是在核心原则上仍然保持一致。

阿玛蒂亚森将功利主义的核心要素归结为三个：“后果主义”“福利主义”“总量排序”，并指出所谓“后果主义”“指的是以下主张：一切选择(无论是对于行动、规则、机构等等所做的)都必须根据其后果(即它们所产生的结果)来评值”[1]。

根据后果论，行为的道德属性取决于其后果的价值，其根本目的在于提高事态的内在价值，而对达到这一目的的手段置之不理。

伯纳德威廉斯指出：“我认为，后果论的中心思想是，只有那种具有内在价值的事物是事态(state of affairs)；任何其他事物具有价值，这是因为它导致了某些具有内在价值的事态。

”[2](8081)也就是说，具有内在价值的事物是事态而非产生价值的行为，行为与行为者只是达成事态的手段。

“功利主义者所感兴趣的，只是幸福的总量。

他们完全不在意幸福是如何产生的，也不在乎是谁的幸福岌岌可危”[3]，这种后果论的理论架构使得功利主义饱受批评。

How to sum up triangles

2
P12P21 ∩ A1B1, C2′ = P12P21 ∩ P23P32, C3′ = P23P32 ∩ A3B3. Let us denote straight lines in an over-determined way by listing those points of the obtained conﬁguration which belong to these lines. Consider the lines
We see that the intersection points of pi and qi give 16 points of our conﬁguration. All of them, except maybe C1′ , C2′ , and C3′ , belong to a curve of degree four that is the union of the four lines r1, r2, r3, and r4. In the notations of the above mentioned theorem from [3] we have C1′ = A43, C2′ = A44, C3′ = A34. Therefore, these points must also belong to the union of the lines ri. It is easy to check that none of the lines ri contains ﬁve points of our conﬁguration. Hence the points C3′ and C1′ lie on r1; and the point C2′ lies on r4. This completes the proof.

小学上册A卷英语第1单元期末试卷(有答案)

小学上册英语第1单元期末试卷(有答案)英语试题一、综合题(本题有100小题，每小题1分，共100分.每小题不选、错误，均不给分)1.The ________ has a loud voice.2.Iron can rust when it comes into contact with _______.3.Astronomy requires careful observation and ______.4.My cousin is a fan of _______ (运动). 她喜欢 _______ (动词).5.She has a ________ for her birthday.6.My favorite ___ (小动物) is the rabbit.7.What is the name of the story about a girl who befriends seven dwarfs?A. CinderellaB. Snow WhiteC. RapunzelD. Aladdin答案: B8.I have a ___ (big) family.9. A ________ (草原) is full of grasses.10. A compound that releases hydrogen ions in solution is called an ______.11.I like building things with my ________ (玩具名). It helps me to be ________ (形容词) and creative.12.What is the name of the main character in "The Great Gatsby"?A. Jay GatsbyB. Nick CarrawayC. Tom BuchananD. Daisy Buchanan 答案: A13.I enjoy picking _____ (野花) in the fields.14.I have a pet ___ (小鸽子) that coos softly.15.The __________ (历史的演变轨迹) provides insights into change.16. A wave can be classified as longitudinal or ______.17. e of Hastings took place in the year ________. The Batt18.I want to learn to ________ (制作手工艺品).19.What is the name of the famous bridge in New York City?A. Brooklyn BridgeB. Golden Gate BridgeC. Tower BridgeD. Sydney Harbour Bridge答案:A20.The dolphin swims in _________. (海洋)21.What do we call a scientist who studies the properties of electricity?A. ElectricianB. PhysicistC. EngineerD. Chemist答案: B22.The ______ loves science fiction.23.The main purpose of proteins is to build and repair _____.24.What is the capital of Ethiopia?A. Addis AbabaB. AsmaraC. DjiboutiD. Nairobi答案:A.Addis Ababa25.The girl is very ________.26.My neighbor, ______ (我的邻居), has a big dog.27.The __________ (历史的工具) aid in research.28.What is the value of 10 5 + 3?A. 5B. 6C. 7D. 8答案:B29. A chemical reaction can produce ______.30.I always carry my _______ (书包) to school.31.My family goes on a ________ (旅行) every summer.32.I want to _______ (建立) a treehouse.33.I have a _____ (collection) of postcards.34.The ________ (气候模式) changes with seasons.35.The bear is a ______ (大型) animal found in forests.36.The owl has a specialized neck that allows it to turn its head almost completely ________________ (旋转).37.My sister enjoys reading ____ (comics) in her free time.38.The _____ (sledge) is ready for winter.39. A __________ can be a solid, liquid, or gas.40.We should _______ (保持)环境整洁。

Lie Symmetries and Solutions of KdV Equation

Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran, I.R.Iran. e-mail: m nadjaﬁkhah@iust.ac.ir † e-mail: reza hejazi@iust.ac.ir
−u
∂ ∂ A2 ux utx + · · · − A5 utx = 0, ∂uxx ∂ux . . . ∂ ∂ − A2 + · · · − A7 = 0, ∂t ∂uxx ∂ ∂ − A4 + · · · + A2 utx = 0. ∂uxx ∂utx
Байду номын сангаас
After solving this system with respect to A1 , ..., A8 , we have A1 = c1 t + c2 , A2 = c1 x/3 + c3 t + c4 , A3 = −2c1 u/3 + c3 , A4 = −c3 ux − 5c1 ut /3, A5 = −c1 ux , A6 = −2c3 utx − 8c1 utt /3, A7 = −c3 uxx − 2c1 utx , A8 = −4c1 uxx /3, where c1 , c2 , c3 , c4 are arbitrary constants, thus v = (c1 t + c2 ) c1 ∂ 2 c1 ∂ 5 c1 ∂ ∂ + ( x + c3 t + c4 ) + (c3 − u) − (c3 u x + ut ) ∂t 3 ∂x 3 ∂u 3 ∂ut 8 c1 ∂ ∂ 4 c1 ∂ ∂ − (2c3 utx + utt ) − (c3 uxx + 2c1 utx ) − uxx . −c1 ux ∂ux 3 ∂utt ∂utx 3 ∂uxx

1 Introduction to the Galois Theory of Linear Differential Equations

∂ ∂ , . . . , ∂x }) = inﬁnitely diﬀerExamples 1.2.2 1) (C ∞ (Rm ), ∆ = { ∂x 1 m m entiable functions on R . ∂ ∂ 2) (C(x1 , . . . , xm ), ∆ = { ∂x , . . . , ∂x }) = ﬁeld of rational functions 1 m ∂ 3) (C[[x]], ∂x ) = ring of formal power series 1 ] C((x)) = quotient ﬁeld of C[[x]] = C[[x]][ x ∂ 4) (C{{x}}, ∂x ) = ring of germs of convergent series 1 C({x}) = quotient ﬁeld of C{{x}} = C{{x}}[ x ] ∂ ∂ 5) (MO , ∆ = { ∂x , . . . , ∂x }) = ﬁeld of functions meromorphic on 1 m open,connected m O ⊂C
1.2 What is a Linear Diﬀerential Equation? I will develop an algebraic setting for the study of linear diﬀerential equations. Although there are many interesting questions concerning diﬀerential equations in characteristic p [MvdP03a, MvdP03b, dP95, dP96], we will restrict ourselves throughout this paper, without further mention, to ﬁelds of characteristic 0. I begin with some basic deﬁnitions.

Notions of symmetry

Broken symmetry revisitedF.A.Bais and M.de Wild PropitiusNotions of symmetrySymmetry has become one of the major guiding principles in physics during the twentieth century.Over the past ten decades,we have gradually progressed from external to internal,from global to local,fromﬁnite to inﬁnite,from ordinary to super symmetry and quite recently arrived at the notion of Hopf algebras or quantum groups.In general,a physical system consists of aﬁnite or inﬁnite number of degrees of freedom which may or may not interact.The dynamics is prescribed by a set of evo-lution equations which follow from varying the action with respect to the diﬀerent degrees of freedom.A symmetry then corresponds to a group of transformations on the space time coordinates and/or the degrees of freedom that leave an action and therefore also the evolution equations invariant.External symmetries have to do with invariances(e.g.Lorentz invariance)under transformations on the space time coordinates.Symmetries not related to transformations of space time coordinates are called internal symmetries.We also discriminate between global symmetries and local symmetries.A global or rigid symmetry transformation is the same throughout space time and usually leads to a conserved quantity.Turning a global symmetry into a local symmetry,i.e.allowing the symmetry transformations to vary continu-ously from one point in space time to another,requires the introduction of additional gauge degrees of freedom mediating a force.It is this so-called gauge principle that has eventually led to the extremely successful standard model of the strong and electro-weak interactions between the elementary particles based on the local gauge group SU(3)×SU(2)×U(1).1Broken symmetryThe use of symmetry considerations has been extended signiﬁcantly by the ob-servation that a symmetry of the action is not automatically a symmetry of the groundstate of a physical system.If the action is invariant under some symmetry group G and the groundstate only under a subgroup H of G,the symmetry group G is said to be spontaneously broken down to H.The symmetry is not completely lost though,for the broken generators of G transform one groundstate into another. The physics of a broken global symmetry is quite diﬀerent from a broken local (gauge)symmetry.The signature of a broken continuous global symmetry group G in a physical system is the occurrence of massless scalar degrees of freedom,the so-called Goldstone bosons.Speciﬁcally,each broken generator of G gives rise to a massless Goldstone bosonﬁeld.Well-known realizations of Goldstone bosons are the long range spin waves in a ferromagnet,in which the rotational symmetry is broken below the Curie temperature through the appearance of spontaneous mag-netization.An application in particle physics is the low energy physics of the strong interactions,where the spontaneous breakdown of(approximate)chiral symmetry leads to(approximately)massless pseudoscalar particles such as the pions.In the case of a broken local(gauge)symmetry,in contrast,the would be massless Goldstone bosons conspire with the massless gaugeﬁelds to form massive vector ﬁelds.This celebrated phenomenon is known as the Higgs mechanism.The canon-ical example in condensed matter physics is the ordinary superconductor.In the phase transition from the normal to the superconducting phase,the U(1)gauge symmetry is spontaneously broken to theﬁnite cyclic group Z2by a condensate of Cooper pairs.This leads to a mass M A for the photonﬁeld in the superconducting medium as witnessed by the Meissner eﬀect:magneticﬁelds are expelled from a superconducting region and have a characteristic penetration depth which in proper units is just the inverse of the photon mass M A.Moreover,the Coulomb interactions among external electric charges in a superconductor are ofﬁnite range∼1/M A.The Higgs mechanism also plays a key role in the uniﬁed theory of weak and electromag-netic interactions,that is,the Glashow-Weinberg-Salam model where the product gauge group SU(2)×U(1)is broken to the U(1)subgroup of electromagnetism.In this context,the massive vector particles correspond to the W and Z bosons medi-ating the short range weak interactions.More speculative applications of the Higgs mechanism are those where the standard model of the strong,weak and electromag-netic interactions is embedded in a grand uniﬁed model with a large simple gauge group.The most ambitious attempts invoke supersymmetry as well.2Topological excitations or defectsIn addition to the aforementioned characteristics in the spectrum of fundamental excitations,there are in general otherﬁngerprints of a broken symmetry in a physical system.These are usually called topological excitations or just defects and correspond to collective degrees of freedom carrying‘charges’or quantum numbers which are conserved for topological reasons,not related to a manifest symmetry of the action. (See,for example,the references[1,2,3,4]for reviews).It is exactly the appearance of these topological charges which renders the corresponding collective excitations stable.Topological excitations may manifest themselves as particle-like,string-like or planar-like objects(solitons),or have to be interpreted as quantum mechanical tunneling processes(instantons).Depending on the model in which they occur, these excitations carry evocative names like kinks,domain walls,vortices,cosmic strings,Alice strings,monopoles,skyrmions,texture,sphalerons and so on.Defects are crucial for a full understanding of the physics of systems with a broken symmetry and lead to a host of rather unexpected and exotic phenomena which are in general of a nonperturbative nature.The prototypical example of a topological defect is the Abrikosov-Nielsen-Olesenﬂux tube in the type II superconductor with broken U(1)gauge symmetry[5,6].The topologically conserved quantum number characterizing these defects is the magnetic ﬂux,which indeed can only take discrete values.A beautiful but unfortunately not yet observed example in particle physics is the’t Hooft-Polyakov monopole[7,8] occurring in any grand uniﬁed model in which a simple gauge group G is broken to a subgroup H containing the electromagnetic U(1)factor.Here,it is the quantized magnetic charge carried by these monopoles that is conserved for topological reasons. In fact,the discovery that these models support magnetic monopoles reconciled the two well-known arguments for the quantization of electric charge,namely Dirac’s argument based on the existence of a magnetic monopole[9]and the obvious fact that the U(1)generator should be compact as it belongs to a larger compact gauge group.An example of a model with a broken global symmetry supporting topological exci-tations is the eﬀective sigma model describing the low energy strong interactions for the mesons.That is,the phase with broken chiral symmetry mentioned before.One may add a topological term and a stabilizing term to the action and obtain a theory that features topological particle-like objects called skyrmions,which have exactly the properties of the baryons.See reference[10]and also[11,12].So,upon extending the eﬀective model for the Goldstone bosons,we recover the complete spectrum of the underlying strong interaction model(quantum chromodynamics)and its low en-3ergy dynamics.Indeed,this picture leads to an attractive phenomenological modelfor baryons.Another area of physics where defects may play a fundamental role is cosmology.See for instance reference[13]for a recent review.According to the standard cosmo-logical hot big bang scenario,the universe cooled down through a sequence of localand/or global symmetry breaking phase transitions in a very early stage.The ques-tion of the actual formation of defects in these phase transitions is of prime impor-tance.It has been argued,for instance,that magnetic monopoles might have beenproduced copiously.As they tend to dominate the mass in the universe,however,magnetic monopoles are notoriously hard to accommodate and if indeed formed,they have to be‘inﬂated away’.Phase transitions that see the production of(localor global)cosmic strings,on the other hand,are much more interesting.In contrastwith magnetic monopoles,the presence of cosmic strings does not lead to cosmo-logical disasters and according to an attractive but still speculative theory cosmicstrings may even have acted as seeds for the formation of galaxies and other largescale structures in the present day universe.Similar symmetry breaking phase transitions are extensively studied in condensedmatter physics.We have already mentioned the transition from the normal to thesuperconducting phase in superconducting materials of type II,which may give riseto the formation of magneticﬂux tubes.In theﬁeld of low temperature physics,there also exists a great body of both theoretical and experimental work on thetransitions from the normal to the many superﬂuid phases of helium-3in whichline and point defects arise in a great variety,e.g.[14].Furthermore,in uniaxialnematic liquid crystals,point defects,line defects and texture arise in the transitionfrom the disordered to the ordered phase in which the rotational global symmetrygroup SO(3)is broken down to the semi-direct product group U(1)×s.d.Z2.Bi-axialnematic crystals,in turn,exhibit a phase transition in which the global rotationalsymmetry group is broken down to the product group Z2×Z2yielding line defects labeled by the elements of the(nonabelian)quaternion group¯D2,e.g.[2].Nematiccrystals are cheap materials and as compared to helium-3,for instance,relativelyeasy to work with in the laboratory.The symmetry breaking phase transitionstypically appear at temperatures that can be reached by a standard kitchen oven,whereas the size of the occurring defects is such that these can be seen by means ofa simple microscope.Hence,these materials form an easily accessible experimentalplayground for the investigation of defect producing phase transitions and as suchmay partly mimic the physics of the early universe in the laboratory.For some recentingenious experimental studies on the formation and the dynamics of topologicaldefects in nematic crystals making use of high speedﬁlm cameras,the interested4reader is referred to[15,16].From a theoretical point of view,many aspects of topological defects have been studied and understood.At the classical level,one may roughly sketch the following programme.Oneﬁrst uses simple topological arguments,usually of the homotopy type,to see whether a given model does exhibit topological charges.Subsequently, one may try to prove the existence of the corresponding classical solutions by func-tional analytic methods or just by explicit construction of particular solutions.On the other hand,one may in many cases determine the dimension of the solution or moduli space and its dependence on the topological charge using index theory. Finally,one may attempt to determine the general solution space more or less ex-plicitly.In this respect,one has been successful in varying degree.For example, the self-dual instanton solutions of Yang-Mills theory on S4have been obtained completely.Properties of a single defectThe physical properties of topological defects can be probed by their interactions with the ordinary particles or excitations in the model.This amounts to inves-tigating(quantum)processes in the background of the defect.In particular,one may calculate the one-loop corrections to the various quantities characterizing the defect,which involves studying theﬂuctuation operator.Here,one naturally has to distinguish the modes with zero eigenvalue from those with nonzero eigenvalues. The nonzero modes generically give rise to the usual renormalization eﬀects,such as mass and coupling constant renormalization.The zero modes,which often arise as a consequence of the global symmetries in the theory,lead to collective coordinates. Their quantization yields a semiclassical description of the spectrum of the theory in a given topological sector,including the external quantum numbers of the soliton such as its energy and momentum and its internal quantum numbers such as its electric charge,e.g.[1,3].In situations where the residual gauge group H is nonabelian,the analysis outlined in the previous paragraph is rather subtle.For instance,the naive expectation that a soliton can carry internal electric charges which form representations of the complete unbroken group H is wrong.As only the subgroup of H which commutes with the topological charge can be globally implemented,these internal charges form representations of this so-called centralizer subgroup.(See[?,19,18]for the case of magnetic monopoles and[20,21]for the case of magnetic vortices).This makes the full spectrum of topological and ordinary quantum numbers in such a broken phase5rather intricate.Also,an important eﬀect on the spectrum and the interactions of a theory with a broken gauge group is caused by the introduction of additional topological terms in the action,such as a nonvanishingθangle in3+1dimensional space time and the Chern-Simons term in2+1dimensions.It has been shown by Witten that in case of a nonvanishingθangle,for example,magnetic monopoles carry electric charges which are shifted by an amount proportional toθ/2πand their magnetic charge[22]. Other results are even more surprising.A broken gauge theory only containing bosonicﬁelds may support topological excitations(dyons),which on the quantum level carry half-integral spin and are fermions,thereby realizing the counterintuitive possibility to make fermions out of bosons[23,24].It has subsequently been argued by Wilczek[25]that in2+1dimensional space time one can even have topologi-cal excitations,namelyﬂux/charge composites,which behave as anyons[26],i.e. particles with fractional spin and quantum statistics interpolating between bosons and fermions.The possibility of anyons in two spatial dimensions is not merely of academic interest,as many systems in condensed matter physics,for example,are eﬀectively described by2+1dimensional models.Indeed,anyons are known to be realized as quasiparticles in fractional quantum Hall systems[27,28].Further,it has been been shown that an ideal gas of electrically charged anyons is supercon-ducting[29,30,31,32].At present it is unclear whether this new and rather exotic type of superconductivity is actually realized in nature.Furthermore,remarkable calculations by’t Hooft revealed a nonperturbative mecha-nism for baryon decay in the standard model through instantons and sphalerons[33]. Afterwards,Rubakov and Callan discovered the phenomenon of baryon decay catal-ysis induced by grand uniﬁed monopoles[34,35].Baryon number violating processes also occur in the vicinity of grand uniﬁed cosmic strings as has been established by Alford,March-Russell and Wilczek[36].Topological interactions between defectsSo far,we have given a(rather incomplete)enumeration of properties and processes that involve the interactions between topological and ordinary excitations.However, the interactions between defects themselves can also be highly nontrivial.Here,one should not only think of ordinary interactions corresponding to the exchange ofﬁeld quanta.Consider,for instance,the case of Alice electrodynamics which occurs if some nonabelian gauge group(e.g.SO(3))is broken to the nonabelian subgroup6U(1)×s.d.Z2,that is,the semi-direct product of the electromagnetic group U(1) and the additional cyclic group Z2whose nontrivial element reverses the sign of the electromagneticﬁelds[37].This model features magnetic monopoles and in addition a magnetic Z2string(the so-called Alice string)with the miraculous property that if a monopole(or an electric charge for that matter)is transported around the string, its charge will change sign.In other words,a particle is converted into its own anti-particle.This nonabelian analogue of the celebrated Aharonov-Bohm eﬀect[39]is of a topological nature.That is,it only depends on the number of times the particle winds around the string and is independent of the distance between the particle and the string.Similar phenomena occur in models in which a continuous gauge group is sponta-neously broken down to someﬁnite subgroup H.The topological defects supported by such a model are string-like in three spatial dimensions and carry a magneticﬂux corresponding to an element h of the residual gauge group H.As these string-like objects trivialize one spatial dimension,we may just as well descend to the plane,for convenience.In this arena,these defects become magnetic vortices,i.e.particle-like objects of characteristic size1/M H with M H the symmetry breaking scale.Be-sides these topological particles,the broken phase features matter charges labeled by the unitary irreducible representationsΓof the residual gauge group H.Since all gaugeﬁelds are massive,there are no ordinary long range interactions among these particles.The remaining long range interactions are topological Aharonov-Bohm interactions.If the residual gauge group H is nonabelian,for instance,the non-abelianﬂuxes h∈H carried by the vortices exhibitﬂux metamorphosis[40].In the process of circumnavigating one vortex with another vortex theirﬂuxes may change. Moreover,if a charge corresponding to some representationΓof H is transported around a vortex carrying the magneticﬂux h∈H,it returns transformed by the matrixΓ(h)assigned to the element h in the representationΓof H.The spontaneously broken2+1dimensional models just mentioned will be the sub-ject of these lecture notes.One of our aims is to show that the long distance physics of such a model is,in fact,governed by a Hopf algebra or quantum group based on the residualﬁnite gauge group H[20,41,42,43].This algebraic framework mani-festly uniﬁes the topological and nontopological quantum numbers as dual aspects of a single symmetry concept.The results obtained are quirte general and strongly suggest that revisiting the symmetry breaking concept in general will reveal similar underlying algebraic structures.7Bibliography[1]S.Coleman,Classical lumps and their quantum descendents,in Aspects ofsymmetry,(Cambridge University Press,Cambridge,1985).[2]N.D.Mermin,Rev.Mod.Phys.51(1979)591.[3]J.Preskill,in Architecture of the fundamental interactions at short distances,edited by P.Ramond and R.Stora,(North-Holland,Amsterdam,1987).[4]R.Rajaraman,Solitons and Instantons,(North-Holland,Amsterdam,1982)[5]A.Abrikosov,JETP5(1959)1174.[6]H.B.Nielsen and P.Olesen,Nucl.Phys.B61(1973)45.[7]G.’t Hooft,Nucl.Phys.B79(1974)276.[8]A.M.Polyakov,JETP Letters20(1974)194.[9]P.A.M.Dirac,Proc.R.Soc.London A133(1931)60.[10]T.H.R.Skyrme,Proc.Roy.Soc.A260(1962)127.[11]D.Finkelstein and J.Rubinstein,J.Math.Phys.9(1968)1762,reprinted inreference[32].[12]E.Witten,Nucl.Phys.B223(1983)433.[13]R.H.Brandenberger,Int.J.Mod.Phys.9(1994)2117.[14]G.E.Volovik,Exotic Properties of Superﬂuid3He,Series in Modern CondensedMatter Physics Volume1,(World Scientiﬁc,Singapore,1992)[15]M.Bowick,L.Chandar,E.Schiﬀand A.Srivastava,Science263(1994)943.8[16]I.Chuang,R.Durrer,N.Turok and B.Yurke,Science251(1991)1336.[17]A.P.Balachandran,F.Lizzi and V.G.Rodgers,Phys.Rev.Lett.52(1984)1818.[18][19]P.Nelson and A.Manohar,Phys.Rev.Lett.50(1983)943.P.Nelson and S.Coleman,Nucl.Phys.B237(1984)1.[20]F.A.Bais,P.van Driel and M.de Wild Propitius,Phys.Lett.B280(1992)63.[21]A.P.Balachandran,F.Lizzi and V.G.Rodgers,Phys.Rev.Lett.52(1984)1818.[22]E.Witten,Phys.Lett.B86(1979)293.[23]G.’t Hooft and P.Hasenfratz,Phys.Rev.Lett.36(1976)1116.[24]R.Jackiw and C.Rebbi,Phys.Rev.Lett.36(1976)1116.[25]F.Wilczek,Phys.Rev.Lett.48(1982)1144.[26]J.M.Leinaas and J.Myrheim,Nuovo Cimento37B(1977)1.[27]B.I.Halperin,Phys.Rev.Lett.52(1984)1583.[28]ughlin,Phys.Rev.Lett.50(1983)1395.[29]Y.-H.Chen,F.Wilczek,E.Witten and B.I.Halperin,Int.J.Mod.Phys.B3(1989)1001.[30]A.Fetter,C.Hanna and ughlin,Phys.Rev.B39(1989)9679.[31]ughlin,Phys.Rev.Lett.60(1988)2677.[32]F.Wilczek editor,Fractional statistics and anyon superconductivity,(WorldScientiﬁc,Singapore,1990).[33]G.’t Hooft,Phys.Rev.Lett.37(1976)8;Phys.Rev.D14(1976)3432.[34]C.Callan,Phys.Rev.D26(1982)2058;Nucl.Phys.B212(1983)391.[35]V.Rubakov,Pis’ma Zh.Eksp.Teor.Fiz.33(1981)658;JETP Lett.33(1981)644;Nucl.Phys.B203(1982)311.9[36]M.G.Alford,J.March-Russell and F.Wilczek,Nucl.Phys.B328(1989)140.[37]A.S.Schwarz,Nucl.Phys.B208(1982)141.[38]S.Shnider and S.Sternberg,Quantumgroups,from coalgebras to Drinfeld al-gebras,a guided tour,International Press Incorporated,Boston,1993).[39]Y.Aharonov and D.Bohm,Phys.Rev.115(1959)485.[40]F.A.Bais,Nucl.Phys.B170(FSI)(1980)32.[41]F.A.Bais,P.van Driel and M.de Wild Propitius,Nucl.Phys.B393(1993)547.[42]F.A.Bais and M.de Wild Propitius,in The III International Conference onMathematical Physics,String Theory and Quantum Gravity,Proceedings of the Conference,Alushta,1993,Theor.Math.Phys.98(1994)509.[43]M.de Wild Propitius,Topological Interactions in Broken Gauge Theories,PhDthesis,Universiteit van Amsterdam,1995,hep-th/9511195.10。

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1paring two images, or an image and a model, is the fundamental operation for many image processing and computer vision systems. In most systems of interest, a simple pixelby-pixel comparison won’t do: the diﬀerence measurement that we determine must bear some correlation with the perceptual diﬀerence between the two images, or with the diﬀerence between two adequate interpretations of the two images. In order to compute meaningful diﬀerences between images, the ﬁrst step is usually the determination of a suitable set of features which encode the characteristics that we intend to measure. Measuring meaningful image similarity is a dichotomy that rests on two elements: ﬁnding the right set of features and endowing the feature space with the right metric. Since the same feature space can be endowed with an inﬁnity of metrics, the two problems are by no means equivalent, nor does the ﬁrst subsume the second. In this paper we consider the problem of measuring distances in feature spaces. In a number of cases, after having selected the right set of features extracted, and having characterized an