Quantum Stability of (2+1)-Spacetimes with Non-Trivial Topology
时间计算方法
Architecture and Performance Methods ofA Knowledge Support System ofUbiquitous Time ComputationYinsheng ZhangInstitute of Scientific & Technical Information of China, Beijing, ChinaCity University of Hong Kong,Hong Kong, ChinaEmail: zhangyinshengnet@Abstract— An architecture and main performance methods of a knowledge support system of ubiquitous time computation based on relativity are proposed. As main results, modern time theories are described as certain relations of term-nodes in a tree, and some space-time computation models in a large scale and time computation models in different time measurement systems (institutions) are programmed as interfaces for time computation in complex conditions such as time-anisotropic movement systems or gravity-anisotropic environments.Index Terms—Space-Time, Relativity, Real Time Communication, Time Ontology, Time MeasurementI.I NTRODUCTIONTime computation is so ubiquitous nowadays, not only in analyzing texts with time terms, but also in real time computation even in circumstance across time zones or in quantum application such as satellite positioning systems, time-anisotropic movement systems, gravity-anisotropic environments, or space scale in the cosmos. As the relativity theory and quantum mechanics, which we call modern time theories, have made great advances, time computation is desirable to be made on the new time knowledge. It is well known that an ontology made up of specific terms in relations can succinctly represent knowledge homogeneously structured in syntactic pattern and stratified in entailments or in contents with stem-branch relations, and easily be applied to navigate knowledge by relational calculus, so a time knowledge support system based on time ontology with some computational models is proposed here to suffice requirement of time computation based on modern time theories.II.E XTENSION OF T IME E XPRESSIONTime mostly is expressed in a form of natural number and suitable for a unified time measure system in the Earth. For example, Dan Ionescu & Cristian Lambiri[1], E.-R.Orderog & H.Dierks[2], and Merlin [3] respectively gave time definitions or expressions for the real-time system, which, however, relativity of time, time computation models which define how to calculate time units, are omitted. In contrast to some software application fields’ research, some time science organizations give serial time expressions based on modern time theories, among which the International Astronomical Union (IAU,1991) made time definition widely accepted in a reality frame [4] . Thus we need to integrate these definitions and expressions in a complete and standard form for ubiquitous time. To do this, we give a time expression as follows.The physical quantity of time can be expressed as a 4-tuple:T=< D,U,M,I > (1) where,D: Data about time in quantity, it may be numbers or circle physical signals indicating time, or symbols expressing a time in quantity; that is, D∈{ time reading, tick, time number expression}.U: Unit, the measure unit such as “second”,” day”.M: Model, the mathematical formulae, using which you get a time quantity by mathematical computations.I: Institution, it may be indicated by a code which stipulates what unit U is meaningful, from which start time point S an interval can be fixed, according to what model M about time can be computed. So we use I( ) to indicate determining a time physical quantity by some parameters.For example, you say “2 seconds”, you might refer to two units of the Universal Time i.e., of coordinated universal time (CUT, or UTC) set by IAU and the finally arbitrated by the International Telecommunication Union (ITU). Of course, you probably might not refer to that, but to an atomic time (AT), as it may. Both the quantities can be computed by the corresponding models issued by the related organizations. Here, the institution determines the meanings of the time as a physical quantity and gives the computation methods, so we can give an expression similar with a programming expression as T=I(D,U,M), here, T serves as a return value ,and I, a function for the other parameters.Clearly, to set up a knowledge support system, we need to consider this time expression, its elements in the tuple will constitute the main profiles.© 2013 ACADEMY PUBLISHER doi:10.4304/jsw.8.11.2947-2955Figure 1. The architecture of the knowledge support system ofubiquitous time computation.III. A RCHITECTURE OF THE K NOWLEDGE S UPPORTS YSTEM We designed such an architecture for the knowledge support system developed by the author for the time computation in the complex systems.The system mainly made up of the 4 components that ①Time Knowledge Navigation, ② Time Measurement and Computation Models, ③ Time Expression Semantics Computation Models,④ Time Institution Knowledge Texts.Component ① accepts users’ requests for knowledge relating to the time measuring data, for example, a user requests for a model for computing the derivation between its time readings and a time unit in another space or in a time measurement system. The kernel of Component ① is a tree describing time knowledge profiles, say its branches are classifications of the time knowledge in certain relations. It is a catalogue of classification and relations of time knowledge, and also mappings between the classification and the knowledge in Component ② and Component ③. It contains institutions I in (1), which determines Component ② and Component ③ in logic, however, Component ② and Component ③ are listed for directing call not through the nodes of institutions.Component ② is the mathematical models for time measurement and computation, written in software programs and can be called for other time computation programs.Component ③ and ④ are discussed in number V and VI.IV. T IME ONTOLOGY.4. 0 General Description sThe tree in Component ① is a time ontology based on modern time theories for logically showing and savingall the knowledge term nodes in certain relations.These relations are potential information for deeper application such as inference based on relational calculus. On time ontology, most studies focus on time expressions and computations of relations between these expressions. For example, Moen’s time ontology is about time concepts in linguistics [5][6].; Frank etc. came up with a plan and principles building space-time in 4 dimensions and 5 tiers [7]. The typical extant time ontology see WordNet in the part of time, DAML time sub-ontology [8],Time Ontology in OWL built by W3C [9] ,and NASASWEET (Semantic Web for Earth an Environmental Terminology)[10]. In addition, ISO 19111 [11] and ISO 19112[12] set out the conceptual schema for spatial references based on geographic identifiers. This work shows various profiles of data structure of time description, yet has the limitations that(1) Time it describes is in the periphery of the Earth, but not in cosmos large scales;(2) The time properties are unraveled only on non-symmetry (non-back as an arrow), a little on relativity, singularity and quantum property.This might lead to difficulties in computations based on modern time theories.In contrast with this work, the time knowledge tree in Component ① is a time ontology based on modern time theories (hereafter “TOboMTT”, the main branches see attachment) .The nodes between any two levels in top-bottom constitute relations which are propositions (note that when we say “A and B in a certain relation”, it just says a proposition) stating the main frame of modern time theories. So, in essence, we have :TOboMTT={N,R }={Propositions} （2）here, N,R refer to nodes and relations respectively.The root (0- level) and the nodes in the next (1-level) are as followingz TimeSpace-Time Type Time Type Time Property Time Measure Time ExpressionThe root “Time” constitutes “has ” relations with the nodes in the 1-level. That is, “Time has the Space-Time Types”, “Time has the Time Types”, “Time has the Time Properties”, “Time has the Time Measures”, “Time has the Time Expressions”. These relations are basic profiles of the up-to-date study on time.The relations of the nodes between the 1 and 2 levels continue such propositions of those relations between 0 and 1 levels, for example, we can say “Time has the Space-Time Types like Euclid Space-Time”, here, “Euclid Space-Time” just is a node in the 2nd level. Thus,© 2013 ACADEMY PUBLISHERthe relations between the 1 and 2 levels are “includes ”, like “Space-Time Type includes Euclid Space-Time”. In the following contexts, we intuitively explain the main nodes which express some important assertions of modern time theories.4. 1 SPACE-Time TYPEAccording to Einstein’s field equation, space andtime are integrated. So we must take space as a parameterof time considering the space-time type. Einstein’s fieldequation see (3) [13]1()+=82R Rg g T αβαβαβαβ−Λπ （3）Here, α and β are space-time dimensions, i.e., α, β=0,1,2,3 and 0 denotes time for the left expression; R αβ is Ricci tensor, it is a 4×4 matrix of the 16 components ofsecond order space-time curvature, R is scalar curvature, g αβ is a 4×4 matrix of metric tensor, Λ is cosmological constant, T αβ is energy-momentum tensor, a 4×4 matrixtoo.From (3), we get (4), i.e., the differentiation of square of space-time intervals:2=ds g dx dy αβαβ (4) here, x,y are curvilineal coordinates, s is space-time interval. (4) adopts Einstein summation convention, normally like in physics, that a repeated index (α or β ) implies summation over all values of that indexed. (3) and (4) are well confirmed by some experiments in the scale 10-13 cm (the radius of a fundamental particle) to 1028 cm (the radius of the universe). A space-time type normally defined by a solution of the equations (3) or (4).See some basic nodes: Space-Time Type Euclidean space-time (absolute time) Riemannian space-time Inertial reference frame space-time Non-inertial reference frame space-time Friedmann- Walke space-time…… If (3) or (4) are determined as the nonlinear partial differential equations about g αβ , we call s is Riemannian space-time, which means space-time is of curvature and might not be flat (flatness is just a special instance, i.e., Minkowski space-time, in which gravity is neglected, it is regarded as inertial). In (3) or (4), if the time in different space places is described as absolutely not different , and independently from its different places and velocities, the space-time is Euclidean space-time or Newton space-time. Friedmann-Lemaître-Robertson-Walker space-time, simply Robertson-Walker space-time [14][15] , put forwarded by Robertson and Walker, and meet the inference of Friedman [16] and Lamaitre [17] , describes homogeneous and isotropic space-time in a non-inertial system, for which, cosmological curvature k and cosmological time t are introduced into (3) or (4). k takes 3 constants 0,1,-1 representing 3 possible space-time types: flatness, positive curvature and negative curvature. If R in (3) is a constant, Robertson-Walker space-time will become some special instance: when R =0, itwill be Minkowski space-time; R >0, de_Sitter space-time;R <0, anti-de_Sitter space-time. Bianchy I space-time is more general than Robertson-Walker that the space-time is homogeneousbut might be anisotropic [18]. Taub-NUT space-time adds magnetic and electric parameters into (3) or (4) [19]. Godel space-time adds rotationally symmetric axis into (3) or (4) [20]. Rindler space-time expresses such space-time determined by inertial system and non-inertial system [21][22]. In some special cases, R is not easy to be determined. To solve (3) or (4), some parameters are given for specialtypes of space-time. These special types include spherical and axial space-time, and time’s elapse may be neglected for a space spot. For (4), Schwarzschild space-time [23] isspherically symmetric beyond a mass sphere. A spherewith great mass and a radius less than Schwarzschild radius is a black hole, which is thought to bear only 3 kinds of information of mass, charge and angular momentum. Schwarzschild black hole is considered as one with only mass, while Ressner-Nordstrom black hole, named as Ressner-Nordstrom space-time, with mass and charge [24][25]; Kerr black hole, named as Kerr space-time with mass and angular momentum [26]; Kerr-Newmanblack hole, named as Kerr-Newman space-time [27], simultaneously have information of mass, charge and angular momentum. Some spherically symmetric space-time like Vaidya space-time [28] and Tolman space-time [29] consider time as the variable of the function of mass and curvature. As an axial metric space-time, Weyl-Levi-Civita space-time [30] is typical. . 4. 2 Time TYPEWhen we solely study time, we can primarily dividetime into the 3 types: Proper time Coordinate time Cosmological time Proper time is the elapsed between two events as measured by a clock that passes through both events. In other words, proper time value is from the real readings of the clock set by an observer in a definite space spot (ifthe measured body moves, then the clock spot and the moved body’s end spot are considered as one area for the two spots are so near for a large scale space). © 2013 ACADEMY PUBLISHERCoordinate time is integrated time under a coordinate system. It is not a real readings for a special spot (the difference between the different spots in the system is neglected), but a stipulated (calculated that it should be) time in the system. Proper time multiplied by (1- v2/c2)-2 is coordinate time (v is the velocity of the body, in which an implied observer is, c is light velocity). If we set a clock in a universe coordinate system indicating the integrated time, it would indicate the universal time (t in Robertson-Walker equation).The proper time in the Earth can be expressed in various forms as the follows.Ephemeris Time (ET) [31] was defined in principle by the orbital motion of the Earth around the Sun. Here, ephemeris is based on Julian calendar which had been reformed to be Gregorian calendar lasted to the nowadays.True solar time (apparent solar time) is given by the daily apparent motion of the true, or observed, Sun. It is based on the apparent solar day, which is the interval between two successive returns of the Sun to the local meridian [32].Mean solar time is the mean values of measured time of the intervals between two Sun passing an identical meridian [33].Sidereal Time is based on a sidereal day; a sidereal day is a time scale that is based on the Earth's rate of rotation measured relative to the fixed stars, normally to the Sun [34]. Sidereal time may be Greenwich Sidereal Time (GST) which calculated by Greenwich Royal Observatory in mean data or Local Sidereal Time (LST) which is computed by adding or subtracting the numbers of timezone [35] .Universal Time (UT) is computed by truly measured time data based on rotation of the Earth, it is a Greenwich Mean Time (GMT) and computed from the start of a midnight of Prime Meridian at Greenwich, and it has different versions such as UT0,UT1,UT2 and Coordinated Universal Time (UTC) for the computations from varying data on non-exact time scales of the Earth rotation. UT0 is Universal Time determined at an observatory by observing the diurnal motion of stars or extragalactic radio sources. It is uncorrected for the displacement of Earth's geographic pole from its rotational pole. This displacement, called polar motion, causes the geographic position of any place on Earth to vary by several metres, and different observatories will find a different value for UT0 at the same moment.UT1 is the principal form of Universal Time. While conceptually it is mean solar time at 0° longitude, precise measurements of the Sun are difficult. UT1R is a smoothly tuned version of UT1, filtering out periodic variations due to tides. UT2 is a smoothed version of UT1, filtering out periodic seasonal variations. UTC is an atomic timescale that approximates UT1. It is the international standard on which civil time is based [36].Atomic time applies the principle of stimulated atom radiation in a constant frequency. The Thirteenth General Conference of Weights and Measures define a second that "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom [37] ". That is a unit of International Atomic Time (ATI).The results of atomic time computed by different local laboratories are called local atomic time.Dynamical Time (DT) [38] is inferred from the observed position of an astronomical object via a theory of its motion, ET is a DT based on revolution of the Earth in replace of UT based on rotation of the Earth meet Newton’s time theory; to meet Einstein’s time theory IAU builds two versions of ET respectively in the system of Terrestrial Dynamic Time (TDT) Barycentric Dynamical Time (TDB).Local civil time is the corrected version of UTC by adding timezone numbers and adjusting daylight saving time [35] .Coordinate time includes centroid coordinate time and Earth-centered coordinate time, they are set by IAU.4. 3 Time propertyThe time properties are divided into 4 kinds as follows.Time PropertyAsymmetryRelativitySingularityQuantum propertyAsymmetry is the property human first discovered, it refers to what seems to be an arrow went out in one direction and not back.Relativity means anisotropy against gravity or in a light-like velocity.Singularity is the property of some places, where the present physical laws break down, or it can be thought of as the property of edge of space-time [39].The quantum property of time refers to that of time in the particle-scale, where time appears the stranger phenomena far from the macro-scale as we see. For example, the former -latter sequence in macro-scale might be isochronous in the quantum –scale [40].4. 4 Time measure4. 4.1 CoordinatorThe space-time expressed in (3) or (4) can’t always be indicated by Cartesian system, mostly due to some properties which are difficult to be indicated by Cartesian system, and also due to the singularity in the space-time which normally cannot be indicated by the real number system. So two kinds of coordinates are mainly introduced, they are general coordinates and special coordinates. The former are popular in common sense, and transforming them for a special purpose we get the latter----special coordinates, which mainly for describing some new metrics -solutions of (3), (4) with some© 2013 ACADEMY PUBLISHERsingularity variables, or for some particular space-time areas.The coordinates special for the metrics are introduced as follows.Schwarzschild coordinate indicates spherical symmetry, it sometimes becomes degeneratation of some more general conditions. Schwarzschild coordinate uses sphere coordinate with the radius r≠2GM/C2 and r≠0, here , G is universal gravitational constant, M is the mass. The coordinate is divided into two areas by r >2GM/C2 and r <2GM/C2 and leads to the two metrics in (3): g00= - (1-2GM/rC2) and g11= (1-2GM/rC2) -1.In Schwarzschild coordinate, there is not the expression that r=2GM/C2 (this is a singularity), but tortoise coordinate covers this singularity.Eddington coordinate does not diverge in r=2GM/C2 and r=0 by the linear transformation of the variables.Kruskal coordinate covers r=2GM/C2 and r=0 too, and more general in indicating space-time than tortoise and Eddington coordinate .Lemaitre coordinate covers r=2GM/C2 with a different method to Kruskal coordinate.Rindler coordinate indicates the space-time determined by both inertial and non-inertial system.Weyl coordinate indicates the function of metric and allows to indicate imaginary numbers.Fermi normal coordinate indicates space-like geodesic which is the trajectory that its covariant differential is 0 for (4). “space-like” denotes the velocity in the area is far less than light speed. And its time axis indicates proper time for a non-inertial or locally inertial conditions.Harmonic coordinate indicates harmonic conditions that coordinates in curved space satisfy a D' Alembert equation, it is a Cartesian-coordinate-like one in curved space.Local inertial coordinate indicates Minkowski space-time.The special coordinates for the particular space-time areas are introduced as follows.Centroid coordinate (center-of-mass coordinate system) is one taking the centre of a space area as the origin of coordinate. These coordinates include non-rotating geocentric reference system, rotating geocentric reference system, Barycentric Celestial Reference System (BCRS), International Celestial Reference System (ICRS).Non-rotating geocentric reference system takes the Earth centre as the origin of coordinate . IAU provides the metric and methods for computing proper time.Rotating geocentric reference system is supposed as rotated with the Earth together, its X3 axis is the rotation axis of the Earth, and it is taken as International Terrestrial Reference System (ITRS) by IAU. For the rotation direction is not considered, the time in non-rotating geocentric reference system and rotating geocentric reference system is the same.Barycentric Celestial Reference System (BCRS) is recommended by IAU, its origin is the mass centre of the solar system,its third axis is approximately the rotation axis of the Earth.International Celestial Reference System is a centroid coordinate, it is made up of circle of right ascension and circle of declination of approximate 600 quasars, the coordinates are provided by International Earth Rotation and Reference Systems Service (IERS) Most general coordinates are introduced by the mathematical textbooks, so they are omitted here.4. 4.2 Measure UNITThe frame of time measure unit is as follows:Measure of timeUnits of measureTime intervalDynamical time intervalDuration fixed time intervalTime interval with the duration fixedby an ephemerisIntegral time scaleDynamical time scale is referred to as measured values of time parameters by physical quantities in a physical system. Basically, a proper time interval is a dynamical time scale.The main units of dynamical time scales in the ontology are concerned with ephemeris time units. A second in ephemeris time is defined as the fraction 1/31,556,925.9747 of the tropical year in Julian calendar for 1900 January 0 at 12 hours ephemeris time by International Committee for Weights and Measures (CIPM), from this unit, Julian century, year, week and day can be worked out.An integral time scale is accumulated value copied from a contracted time start point, for example, atomic time scale. So it may be proper time or coordinate time.V. T IME MEASURE AND COMPUTATION MODELSComponent ②is the set of the measure and computation models, which are from two resources: one is from the institutions put forward by some organizations such as IAU stipulating how to measure and computation, another resource is from the exact solutions of the (3) or (4).The models are programmed in Mathematica as the Application Programming Interface (API) so that a users’ programs can call these API.EXAMPLE 1[41]: a model (group) to compute a coordinated universal timeUTC (t) – TAI(t) = ns (5)UTC (t) –UT 1(t)=<0.9s; (6) Here, UTC(t) is a time expressed in coordinated universal time’ institution unit, TAI(t) means a time of Atomic Time International, n is natural number; s is the second, UT 1(t) is a time expressed in UT 1.© 2013 ACADEMY PUBLISHEREXAMPLE 2 is calling from a user’s application for the interface of a model, which is drawn from reference [42] and re-wrote by the author, to get an exact solution of Einstein’s field equation given Roberson-Walker Metric:1 /*An application from users in pseudo-code callingthe model-interface. See the tree in the attachment*/2 e num Space-Time in non- inertial system3 {4 B ianchi I Space-Time,5……6R obertson-WalkerSpace-Time7 /*Here, all the 16 Space-Time in non- inertialsystem in the tree enumerated */8 } Metric[16];9 for(i=0;i<16;i++){10 switch(Metric [i])11 case Robertson-Walker Space-Time:12 input and assign vector:13 v = {t, r, e, phi};141516 M = {-1, R[t]^2/(1 - K (r^2), (r^2) (R [t]^2), (r^2) (Sin[e]^2) (R [t]^2)};Call Einstein [M, v]}“Einstein.m”1 E instein [g_, v_] := Block[2 {invsg, dg1, dg2, dg3, Christf2, dChristf2, Ruv1,Ruv2, Ruv3, Ruv4, RicciTensor, R, EMTensor} 3 EMTensor = {}; (*Save return value.*)(*Calculate the inverse metric of g.*)4 g=DiagonalMatrix[M];5 invsg = Inverse[g];(*Calculate the affine connection.*)6 dg1 = Outer[D, g, v];7 dg2 = Transpose[dg1, {1, 3, 2}];8 dg3 = Transpose[dg1, {2, 3, 1}];9 Christf2 = (1/2) invsg.(dg1 + dg2 - dg3);(*Calculate the Ricci tensor.*)10 dChristf2 = Outer[D, Christf2, v];11 Ruv1 = Table[Sum[dChristf2[[k, i, k, j]], {k, 4}],{i, 4}, {j, 4}];12 Ruv2 = Table[Sum[dChristf2[[k, i, j, k]], {k, 4}],{i, 4}, {j, 4}];13 R uv3 = Table[Sum[Christf2[[k, i, j]] Christf2[[h,k, h]], {k, 4}, {h, 4}], {i, 4}, {j, 4}];14 Ruv4 = Table[ Sum[Christf2[[k, i, h]] Christf2[[h,j, k]], {k, 4}, {h, 4}], {i, 4}, {j, 4}];15 RicciTensor = Ruv1 - Ruv2 - Ruv3 + Ruv4;(*Calculate the Curvature Scalar.*)16R = Sum[invsg[[i, i]] RicciTensor[[i, i]], {i, 4}];(*Calculate the field equation left part.*)17EMTensor = RicciTensor - (1/2) g R ;18return [EMTensor]19]20End[]21EndPackage[]This program is divided into two parts: the first part is user’s input for computation, which is space-time dimensions v in a spherical coordinator, in which, t is the cosmological time (see 4. 2 Time Type), M is Roberson-Walker Metric. Users can input similar metrics for calling the function Einstein[ ],which is saved in the second part, a document Einstein.m, starting from the sentence BeginPackage["Einstein`"]. mathlink.h in VC++ enables to run Mathematica programs in VC++ environment The section Block[] is a function of local variables for calling.Outer[] is to give the partial derivative ∂f/∂x.Transpose[dg1, {1, 3, 2}] is to transposes dg1 so that the k th level in dg1 is the n k th level in the result.D [] is to get partial differential.Table [] is to generate a list of the expression Sum[].Sum[] is to get sum.The line 19 is the computation result of left part of (3), yet the cosmological constant is omitted. The right part of (3) is considered as zero.VI. M ECHANISM AND R UNNING OF T HE A RCHITECTURETOboMTT is designed to be a tree not only for satisfying the structure and classification of knowledge of time, but also for developing the knowledge in Web Ontology Language (OWL), which is based on Resource Description Framework (RDF) in a tree. Thus we can divide TOboMTT into some sub-trees and further expressed them in OWL or RDF. Figure 2 is a sample of Class—SubClass relation in RDF. As a result, navigation of knowledge of time, based on TOboMTT, become navigation of resources and serves, based on eXtensible Markup Language (XML) compatible with both OWL and RDF.A query for a sub-class or property value will give the corresponding answer by rational calculus on a XML scheme. For the example in Figure 2, “Space-Time Type includes Euclid Space-Time ” will be the answer for the query “What kind does the Space-Time Type include?” Therefore, query and answer is the first and direct results of navigation of knowledge of time by TOboMTT.<?xml version="1.0"?>© 2013 ACADEMY PUBLISHER。
时滞可交换四元数神经网络稳定性分析
2 准备工作
2.1 交换四元数 一个交换四元数可以表示为：
q = qR + qIi + qJ j + qKk ∈
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智能科学与技术学报
第2卷
q(s) = φ(s) ∈ n , s ∈[t0 −τ , t0 ] φ(s) = φR (s) + φI (s)i + φJ (s) j + φK (s)k
其中，t0 为式（3）表示的系统的初始时间，τ 为时 滞项τ (t) 的上界， φ(s) 为系统初始条件。
假设 1 设 q(t) = qR (t) + qI (t)i + qJ (t) j + qK (t)k,
考虑如下的四元数神经网络：
q(t) = −Cq(t) + Af (q(t)) + Bg(q(t −τ (t))) + u （3）
其中， q(t) = (q1(t), q2 (t), , qn (t))T ∈ n 是具有 n 个 神 经 元 的 CQVNN 在 t 时 刻 的 状 态 向 量 。 C = diag{c1, c2, , cn}∈ n , ci > 0 (i = 1, 2, , n) ，是 自反馈连接矩阵， A = AR + AI i + AJ j + AK k ∈ n×n
注 1 此规则定义的四元数乘法虽然满足了乘 法交换律，但由此衍生出来的交换四元数的模不满 足三角不等式。例如：
时间尺度上相空间中非Chetaev型非完整系统的Noether理论
时间尺度上相空间中非Chetaev型非完整系统的Noether理论祖启航;朱建青;宋传静【摘要】研究了时间尺度上相空间中非Chetaev型非完整力学系统的Noether 理论.首先,基于Hamilton原理,建立了时间尺度上非Chetaev型非完整力学系统的Hamilton方程;其次,根据时间尺度上Hamilton作用量在无限小变换下的广义准不变量,得到了时间尺度上相空间中非Chetaev型非完整力学系统的Noether 等式和守恒量;最后,举例说明结果的应用.【期刊名称】《华中师范大学学报（自然科学版）》【年(卷),期】2017(051)001【总页数】5页(P23-27)【关键词】时间尺度;相空间;非完整系统;Noether等式;守恒量【作者】祖启航;朱建青;宋传静【作者单位】苏州科技大学数理学院,江苏苏州215009;苏州科技大学数理学院,江苏苏州215009;南京理工大学理学院,江苏南京210094【正文语种】中文【中图分类】O3161988年德国学者Hilger在他的博士论文[1]中提出测度链上的微积分理论，其主要思想就是把连续和离散进行统一[2-3].时间尺度作为测度链的一种特殊形式，非常具有代表性.目前，时间尺度在动态方程、变分原理、最优控制和经济等相关领域都得到了广泛的应用[4-11]．近年来，国内外学者对时间尺度上力学系统的变分问题及其对称性与守恒量进行了研究.Bohner研究了时间尺度上Lagrange方程表达形式及变分问题[12]，Barosiewicz等研究了时间尺度上Lagrange系统的Noether理论[13]，Cai等研究了时间尺度上非保守和非完整力学系统的Noether理论[14]，Song和Zhang建立了时间尺度上Birkhoff方程，给出了Birkhoff系统的Noether等式与守恒量[15].本文基于时间尺度上Hamilton原理，建立了时间尺度上非Chetaev型非完整力学系统的Hamilton方程.根据Hamilton作用量在无限小变换下的准不变量，得到了系统的Noether定理．时间尺度上的微积分理论可参阅文献[6]．假设力学系统的位形由n个广义坐标来确定，其运动受时间尺度上g个双面理想非Chetaev型非完整约束非完整约束(1)加在虚位移上的限制条件为时间尺度上Lagrange函数为则有时间尺度上Lagrange非完整力学系统的微分方程[14]其中,为非势广义力，λβ是约束乘子.假设系统非奇异，即对约束条件(1)求Δ导数，并将方程(4)显示形式表示出来[7],.由(6)式解得代入(7)式，则可解出约束乘子λβ作为t，qσ，qΔ的函数.方程(4)可表示为其中，引进时间尺度上广义动量和Hamilton函数[9]于是在正则变量p，qσ下，(1)、(2)和(9)式变为时间尺度上非保守力学系统的Hamilton原理为其中,，满足以下交换关系和端点条件将(13)式两边同时乘以，代入(15)式，可得对(11)式两边关于广义动量求偏导数，得到将(19)式代入(18)式，根据Dubois-Reymond定理[12]，可得对(20)式求Δ导数，可得方程(19)和(21)称为时间尺度上相空间中非Chetaev型非完整力学系统的运动方程.由(14)式，方程(19)和(21)可进一步表示为称方程(22)为与时间尺度上相空间中非完整系统(12)，(19)和(21)相应时间尺度上相空间中完整系统的运动方程．首先，考虑只含有qs，ps变分的情况.相空间中Hamilton作用量表示为定义1称作用量(23)式在变换下为广义准对称不变量，当且仅当对任意区间[ta，tb]⊆[t1，t2],有，其中，ε为无限小参数，ξs和ηs为无限小变换的生成函数，为全变分，为规范函数并且有G=εG．定理1如果作用量(23)式是变换(24)式下的广义准对称不变量，对所有，那么.证明由定义1，方程(25)在任意区间[ta，tb]⊆[t1，t2]上均成立，则(25)式等价于，对(27)式两边同时关于ε求偏导数并令ε=0，则可以得到(26)式．定理2如果作用量I是定义1下的广义准对称不变量，那么系统的守恒量为证明由(22)和(26)式，可得于是得到(28)式.下面将讨论含时间t的无限小变换下的广义准对称不变量．令U是右稠连续可微函数和的集合.对任意qs，ps∈U和ε，映射∈是右稠连续的，而且它是在新的时间尺度上带有前跳算子σ*和导数Δ*的一个象.同时有交换关系[6]:定义2如果作用量I是变换(30)式下的广义准对称不变量，当且仅当对任意的区间[ta，tb]⊆[t1，t2].t.定理3如果作用量I是变换(30)式下的广义准对称不变量，那么.证明由定义2，可得，由于区间[ta，tb]是[t1，t2]的任意子区间，所以有，对(34)式两边同时关于ε求偏导数并令ε=0，则可得等式(32)．(32)式就称为时间尺度上相空间中非Chetaev型非完整力学系统的Noether等式．定理4如果作用量I是定义2下的广义准对称不变量，那么系统的守恒量为证明令，当时，则根据等式(33)有，t.由于=t，则有，).由定义1可知，泛函是在={}上的无限小变换的准不变量.因此当=t，由定理2可得.又因为，其中，∂1H表示对函数H中第一个变量求偏导数.将(41)、(42)式代入(40)式，则可得(35)式．定理4称为时间尺度上相空间中非Chetaev型非完整系统的广义Noether定理，根据这个定理可由已知的广义准对称不变量得到系统的守恒量．定义时间尺度，假设系统的Lagrange函数为所受的非完整约束为该约束为非Chetaev型的，虚位移满足根据(10)式和(11)式，有广义动量和Hamilton函数，将Hamilton函数代入(21)式，则有由(44)，(46)和(47)式，求得于是有根据(32)式和(2)式，可得对(50)和(51)式进行求解所以根据定理4，可得到守恒量时间尺度将离散和连续进行了统一，研究时间尺度在分析力学的应用并寻求相应的守恒量.本文通过时间尺度上Hamilton原理，建立时间尺度上非Chetaev型非完整Hamilton方程.定义了时间尺度上相空间中的广义准不变量，得到系统的Noether等式和守恒量.本文结果具有普遍性，当约束条件时，结论可退化为时间尺度上相空间中Chetaev型非完整力学系统的Noether的理论.同时，可进一步拓展到时间尺度最优控制，约束Birkhoff力学系统等．致谢：作者对张毅教授的悉心指导深表感谢!【相关文献】[1] HILGER S. Ein maβkettenkalkul mit anwendung auf zentrumsmannigfaltigkeiten[D]. Wurzburg:Universität Wurzburg， 1988．[2] HILGER S. Analysis on measure chains-a unified approach to continuous and discrete calculus[J]. Results Math， 1990， 18(1-2):18-56．[3] HILGER S. Differential and difference calculus-unified[J]. Nonlinear Anal， 1997，30(5):2683-2694．[4] AGARWAL R P， BOHNER M. Basic calculus on time scales and some of its applications[J]. Results Math， 1999， 35(1-2):3-22．[5] AGARWAL R P， BOHNER M， PETERSON A. Inequalities on time scales: a survey [J]. J Math Inequ Appl， 2001， 4(4):535-557．[6] BOHNER M， PETERSON A. Dynamic equations on time scales， An Introduction with applications[M]. Boston: Birkhäuser， 2001．[7] BOHNER M， GUSEINOV G SH. Partial differentiation on time scales[J]. Dyn Syst Appl，2004， 13(3): 351-379．[8] ATICI F M， BILES D C， LEBEDINSKY A. An application of time scales to economics[J]. Math Comput Model， 2006， 43(7-8): 718-726．[9] AHLBRANDDT C D， BOHNER M， RIEDNHOUR J. Hamiltonian systems on time scales[J]. J Math Appl Anal， 2000， 250(2): 561-578．[10] HILSCHER R， ZEIDAN V. Weak maximum principle and accessory problem for control problems on time scales[J]. Nonlinear Anal， 2009， 70(9):3209-3226．[11] HILSCHER R， ZEIDAN V. Calculus of variations on time scales: Weak local piecewise solutions with variable endpoints[J]. J. Math Anal Appl， 2004， 289(1):143-166．[12] BOHNER M. Calculus of variations on time scales[J]. Dyn Syst Appl， 2004，13(12):339-349．[13] BARTOSIEWICZ Z， TORRES D F M. Noether theorem on time scales[J]. J Math Anal Appl， 2007， 342(2): 1220-1226．[14] CAI P P， FU J L， GUO Y X. Noether symmetries of the nonconservative and nonholonomic system on time scales[J]. Sci China: Phys Mech Astron， 2013，56(5):1017-1028．[15] SONG C J， ZHANG Y. Noether theorem for Birkhoffian systems on time scales[J]. J Math Phys， 2015， 56(10): 102701(1-7)．。
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仿射对称空间SU(1，2)
第 63 卷第 1 期2024 年 1 月Vol.63 No.1Jan.2024中山大学学报（自然科学版）（中英文）ACTA SCIENTIARUM NATURALIUM UNIVERSITATIS SUNYATSENI仿射对称空间SU(1，2)/SO(1，2)上的Plancherel定理*金四海，范兴亚新疆大学数学与系统科学学院，新疆乌鲁木齐 830017摘要：研究了Hilbert空间L2(Z，μ)上酉表示的不可约分解，其中Z=SU(1，2)/SO(1，2)是Hermitian型仿射对称空间，μ是群SU(1，2)作用在Z上不变的Haar测度. 利用SO(1，2)不变的分布函数，具体的构造了缠结算子，进而得到了L2(Z，μ)上的离散序列表示. 在此基础上，结合离散序列表示的正交补部分，证明了L2(Z，μ)上的Plancherel公式.关键词：仿射对称空间；离散序列表示；Plancherel定理中图分类号：O152.5 文献标志码：A 文章编号：2097 - 0137（2024）01 - 0173 - 08 Plancherel theorem for the affine symmetric space SU(1，2)/SO(1，2)JIN Sihai， FAN XingyaCollege of Mathematics and System Sciences，Xinjiang University，Urumqi 830017， ChinaAbstract：The irreducible decomposition of unitary representations is investigated on the Hilbert spaceL2(Z，μ)，where Z=SU(1，2)/SO(1，2)，and μdenotes an SU(1，2)-invariant Haar measure on Z.By using the SO(1，2)-invariant distribution functions， the intertwining operators is constructed concrete‐ly， and then the discrete series representations on L2(Z，μ) are obtained. On this basis， combined with the orthogonal complement parts of the discrete series representations，the Plancherel formula onL2(Z，μ) is proved.Key words：affine symmetric space； discrete series； Plancherel theorem李群表示理论是近现代分析学的主要研究领域之一，隶属于抽象调和分析范畴，是Fourier级数和Fourier变换理论的深远推广，此方面的研究成果极大地推动了数学物理等相关学科的发展. 悉知，酉表示中不可约表示的分解是李群表示理论研究的主要问题之一，目的是给出抽象的Plancherel公式，进而构造缠结算子来解决相关实际问题. 在此分解中，离散谱的分解是一种极端的情况，此情况可以利用分析的工具来处理. 基于此想法，本文在Hermitian型仿射对称空间上考虑其离散谱的具体构造. 从几何结构方面来讲，Hermitian型仿射对称空间是伪黎曼对称空间，此空间隶属于半单对称空间，据此赋予了Hermite结构的齐性空间. 20世纪80年代初，Flensted-Jensen（1980）系统地研究了半单对称空间上的离散序列表示，并得到离散序列表示存在的必要条件. Flensted-Jensen的这一工作引起了许多表示论专家的关注，究其原因是此理论极大地继承并发展了Harish-Chandra的经典离散序列表示（Harish-Chandra，1965）.近年来，Hermitian型仿射对称空间上的离散序列表示受到了国内外广大学者的关注. 该领域主要关心的是Gelfand-Gindikin问题（Gelfand et al.，1977）. 最近，Delorme et al.（2021）研究了扭曲离散序列表示，此DOI：10.13471/ki.acta.snus.2022A091*收稿日期：2022 − 10 − 19 录用日期：2023 − 02 − 20 网络首发日期：2023 − 11 − 15基金项目：国家自然科学基金（12161083）；国家自然科学基金天元数学访问学者项目（12126360）；新疆维吾尔自治区自然科学基金（2020D01C048，2021D01C071）作者简介：金四海（1997年生），男；研究方向：李群表示论；E-mail：****************通信作者：范兴亚（1986年生），男；研究方向：李群表示论；E-mail：*****************.cn第 63 卷中山大学学报（自然科学版）（中英文）离散序列表示旨在给出Gelfand-Gindikin 问题的回答，主要的方法是通过Bersentan 态射来构造Plancherel 公式. 此方法简化了离散序列表示与主序列表示的复杂计算，主要判别工具是限制球根系是否为零向量空间，进而给出抽象的Plancherel 公式. 从理论方面来讲，此方法值得推广（韩威等，2023）. 但是在Hermite 情形下，Delorme et al.（2021）的方法不能具体的构造扭曲离散序列表示，究其原因是限制球根系对应的极大交换子空间是一个零向量空间. 至于0-扭曲离散序列表示，可以通过'Olafsson et al.（1988）的缠结算子来构造此序列.根据Flensted-Jensen （1980）提出的必要条件，容易判定Hermitian 对称空间SU ()1，2/SO ()1，2上存在离散序列表示. 但没有给出具体的形式，因此有必要去研究它的具体构造. 基于'Olafsson et al.（1988）的思想，本文具体地构造缠结算子，进而展开研究SU ()1，2/SO ()1，2上的离散序列表示，并在限制球根系上，找到了判别离散序列表示存在的一个新的等价条件. 在Plancherel 公式中Plancherel 测度的计算是一个核心问题. 在Duistermaat et al.（1979）的基础上，本文精确地计算了此具体情形下的Plancherel 测度，并得到了对应的Plancherel 公式.1 符号准备及主要引理设SU ()1，2是SL ()3，C 的子群，此群保不定内积[]z ，z =||z 12-||z 22-||z 32，其中z =(z 1，z 2，z 3)∈C 3.群SU ()1，2对应的李代数为su ()1，2. 定义su ()1，2上的Cartan 对合为θ(X )=-X *，其中X *表示X 的共轭转置. 在此对合意义下，su ()1，2有Cartan 分解k⊕p ，其中k ={}X ∈su ()1，2 | θ()X =X ，p ={}X ∈su ()1，2 | θ()X =-X . 此外，在su ()1，2上引进另外一种对合τ(X )=X ，其中X 为X 的共轭，则有相应的τ分解h⊕q ，其中h ={}X ∈su ()1，2 | τ()X =X ，q ={}X ∈su ()1，2 | τ()X =-X .易证θτ=τθ，结合此对合交换关系，得到su ()1，2关于θτ的正交分解su(1，2)= (k ∩h)⊕(p ∩q)g +⊕(h ∩p)⊕(k ∩q)g -=g +⊕g -.易证g +是su ()1，2的子空间，且对su ()1，2上的李括积运算封闭，则g +是su ()1，2的李子代数. 设a p ∩q 是p ∩q 的极大交换子空间，记作a ≔a p ∩q =ìíîïïï|||||||i ()0t 0-t000 t ∈R üýþïïïï.注意，此处a 的取法还有另外一种形式ìíîïïï|||||||i ()00t000-t t ∈R üýþïïïï.此取法在下文计算根系时，所得根系与上面对应的结果一致，故此全文仅考虑第一种情形.利用a 在su ()1，2上的伴随作用，可得根系±2α与±α所对应的特征向量分别如下()i 101-i 0000， ()i -10-1-i 0000， ()00±i 00-1∓i10，()00±i 001∓i -10， ()0100i 1i0， ()00100-i 1-i 0.令a *是a 的对偶空间，定义SU ()1，2在a 下的根系为Σ(su(1，2)，a)={±2α，±α}，其中α∈a *且α(X )=t ，X ∈a . Σ(su(1，2)，a)的正根系为Σ+(su(1，2)，a)={}2α，α. 由根系的反射可得Weyl 群，此群同构于对称群S 2.定义1 设R 是由{}2α，α生成的半群，且R ⊂a */{0}. 设S ={α}⊂R ，定义压缩锥为a -≔{X ∈a |∀γ∈S ，γ(X )≤0}.注 1 设a E ={X ∈a |∀γ∈R ，γ(X )=0}，利用定义1，可知a E =a -∩(-a -)≡{0}. 值得一提的是，174第 1 期金四海，等：仿射对称空间SU(1，2)/SO(1，2)上的Plancherel 定理a E ≡{0}为仿射对称空间SU ()1，2/SO ()1，2上存在离散序列表示的必要条件，这是Harish-Chandra 定理（离散序列表示存在性定理）的推广. 事实上，设a=a +a E =a. 利用条件a E ={0}，可知商空间a /a 没有非平凡对偶，从而SU ()1，2/SO ()1，2中存在离散序列表示. 这种离散序列表示称为0-扭曲离散序列表示（Krötz et al.，2020），此0-扭曲离散序列表示恰好对应的是SU ()1，2/SO ()1，2中的离散序列表示，具体见文献（Krötz et al.，2020）. 结合Flensted-Jensen （1980）的定理1.1，只需验证条件：rank(SU(1，2)/SO(1，2))=rank ()SO(3)/()SO(3)∩SO(1，2).此条件可根据SU ()1，2的极大环面的维数证实，从而SU ()1，2/SO ()1，2满足Flensted-Jensen 条件.引理1 g +的根系为Σ()g +，a ={±α}，其对应的Weyl 群同构于对称群S 2.证明 由k ，h ，p 和q 的定义，可知k ∩h =ìíîïïï|||||||()0000k 230-k 230 k 23∈R üýþïïïï， p ∩q =ìíîïïïi |||||||()0x 1x 2-x 100-x 2x 1，x 2∈R üýþïïïï.于是g +=(h ∩k)⊕(p ∩q)≔ìíîïïï|||||||()0i x 1i x 2-i x 10a 23-i x 2-a 23x 1，x 2，a 23∈R üýþïïïï.令h =i ()10-1000， 则方程[]X ，h =±X 对应的特征向量为()00±i 00-1∓i10，()00±i001∓i -10的常数倍. 利用上式，结合g +的定义，可知Σ()g +，a ={±α}. g +的Weyl 群由根系Σ()g +，a ={±α}的反射组成，此群同构于对称群S 2，引理证毕.注2 利用引理1，得到Σ()g +，a 的素根系为I ={α}. 定义a I ≔{X ∈a | α(X )=0}，易证a I ={0}，此条件暗示了在限制根系意义下，SU ()1，2/SO ()1，2上不存在非零扭曲离散序列表示.2 主要结果设n =n ()Σ+()su ()1，2，a ，定义P =LN 为SU ()1，2中τθ不变的极小抛物子群，其中N =exp (n )，L 是a 在SU ()1，2上的中心化子，即L =ìíîïïï|||||||()l 1l 20-l 2l 1000-1l 1，l 2∈C ，||l 12-||l 22=1üýþïïïï.设L =MA ，其中M 是τ稳定子群，A =exp (a)=ìíîïïïï|||||()cosh (t )i sinh (t )0-i sinh (t )cosh (t )0001 t ∈R üýþïïïï.定义2 子群P 诱导SU ()1，2的主序列表示定义为：H σ，λ=Ind SU(1，2)P(σ⊗λ)={f | f (g ，m ，a ，n )=σ(m )-1(a )-λ+ρf (g )，g ∈G ，m ⋅a ⋅n ∈P }，其中ρ为正根和的一半，σ是M 的离散序列表示，λ∈ia *.现具体构造M 的离散序列表示，其记号为()σ，V σ.175第 63 卷中山大学学报（自然科学版）（中英文）引理2 M /(M ∩H )≃SU ()1,1/SO ()1,1，其中H =SO ()1，2 .证明 由于M 是L 的τ稳定子群，则M ≃SU ()1，1. 设H =ìíîüýþ|||()db t cA ∈SL (3，R ) b ，c ∈R 2，d ∈R ，A t A -c t c =1，A t b =dc ，b 2-d 2=-1，从而M ∩H =ìíîïïï|||||||()d b 0-bd 000-1b ，d ∈R ，d 2-b 2=1üýþïïïï≃SO(1，1).引理证毕.利用Harish-Chandra 嵌入，可知SU ()1，1/SO(2)≃D ={}z ∈C | 1-||z 2>0 . 假定ω∈R ，考虑加权平方可积空间L 2，ω(D )：L 2，ω(D )=ìíîïïF (z ) |||| F 2L 2，ω(D )=c ω∫D ||F (z )2[1-||z 2]ω-2d z <+∞üýþïï，其中c ω是一个亚纯系数，且满足 1L 2，ω(D )=1，d z 是C 上的Lebesgue 测度. 定义L 2，ω(D )在群SU ()1，1作用下的酉表示为[πω(g )f ](z )=()b z +a -ω-2f (g -1⋅z )，其中g -1=()ab b a ∈SU ()1，1，且g -1⋅z =(az +b )()b z +a -1.当n =ω≥2且n ∈Z +时，在L 2，ω(D )上定义一个加权Bergman 空间H n (D )≔ìíîïïf ∈O (D )|||| f 2L 2，ω(D )≔n -1π∫D ||f (z )2()1-||z 2n -2d z <+∞üýþïï，其中O (D )是D 上全纯函数构成的集合. 易证πn 是SU ()1，1在H n (D )上的全纯离散序列表示. 对于任意的g ∈SU ()1，1，设H ∞n (D )是πn (g )作用在H n (D )上关于g 的无穷可微的函数构成的空间. 定义H -∞n (D )是H ∞n (D )的前对偶空间，根据空间的嵌入关系可知H ∞n (D )⊂H n (D )⊂H -∞n (D ).引理3（'Olafsson et al.，1988） 设n ≥2且n ∈Z +，定义π∨n 为πn 的逆步表示. 设x 0=eH ，g =k θa t x 0，其中k θ∈K ，a t ∈A . 对于任意的h (z )∈H ∞n (D )，则存在一个SO ()1，1不变的分布Φn ∈H -∞n (D )使得缠结算子I n h (z )(g )=h ，π∨n (g )Φn 可将H n (D )等距地延拓到V n ，其中V n 是L2()SU ()1, 1/SO ()1, 1的闭子空间.当n ≥2且n ∈Z +时，则π-n 为SU ()1，1的共轭全纯离散系列表示，其表示空间为闭子空间V -n . 结合引理2和引理3，可知M 的离散序列表示有同构关系σ≃π±n ，记作σ±n .为了得到本文的主要定理，首先需要计算Plancherel 测度. 由于{}α，2α为a *中的一组基，于是对于∀β∈a *，β=a 1(α)+a 2(2α)，其中a 1，a 2∈R . 令a 1+2a 2=λ1，将a *中的元素记作λ=(λ1，0).命题 1 ia *上的Plancherel 测度为d λ=||||||||||27Γ(λ1)Γ()12+λ12Γ()52Γ()748Γ(1+λ1)Γ()1+λ12Γ()32Γ()54||||||||||-2d λ1.证明 结合文献（Duistermaat et al.，1979），可知ia *上的Plancherel 测度为d μ(λ)=||c (λ)-2d λ.（1）设ρ为Σ()g ，a 的半根系，其中c (λ)=I ()λI ()ρ， I (λ)=I (α；λ).176第 1 期金四海，等：仿射对称空间SU(1，2)/SO(1，2)上的Plancherel 定理利用正特征向量的重数，得到n (α)=dim (g α)=2， n (2α)=dim (g 2α)=1， d (α)=n (α)+n (2α)=3，从而I (α；λ)=α，α-12d (α)Γ()λ，αα，αΓ()14n (α)+12λ，αα，αΓ()12n (α)+λ，αα，αΓ()12n (2α)+14n (α)+12λ，αα，α，其中Γ(⋅)定义为Gamma 函数，⋅，⋅为a *中的Killing 形式. 代入λ和ρ得I (α；λ)=Γ(λ1)Γ()12+λ12Γ(1+λ1)Γ()1+λ12， I (ρ；λ)=8Γ()32Γ()5427Γ()52Γ()74.于是c (λ)=27Γ(λ1)Γ()12+λ12Γ()52Γ()748Γ(1+λ1)Γ()1+λ12Γ()32Γ()54.现将c (λ)代入式（1），此命题得证.定理1（Plancherel 公式） 仿射对称空间Z =SU ()1, 2/SO ()1, 2上的Plancherel 公式为L 2(Z )=∑n ≥2∫ia *⊕H σ±n， λd λ⊕∑||ν-22<r ≤||ν-2F 2r +4-||ν，其中σ±n 和d λ如上所述，F 2r +4-||ν见下文.3 Plancherel 公式的证明首先构造定理1中的离散序列表示部分. 本部分的构造需要一些准备和技巧. 提醒读者的是，如下所定义的符号和第2节定义的单位圆盘不同，因此空间的定义域和指标也不尽相同.定义B ≔{}z =(z 1，z 2)∈C 2 |||z 12+||z 22<1. 悉知，B 的自同构群为SU ()1, 2，且SU ()1, 2通过分式线性变换可迁地作用在B 上. 定义Hilbert 空间L 2(B ，d μν(z ))=ìíîïïf |||| ∫B ||f (z )2()1-||z 2||ν-3d z <+∞üýþïï，其中d z 是Lebesgue 测度，ν∈R . 如果|ν|>2，||ν∈Z +，定义Hilbert 空间H ν(B )=ìíîïïf 是全纯函数 |||| f 2=∫B ||f (z )2()1-||z 2||ν-3d z <+∞üýþïï.特别地，如果||ν=3，那么H 3(B )是经典的Bergman 空间. 定义群SU ()1, 2在H ν(B )上的酉表示为[πν(g )f ](z )=[det (cz +d )]-||νf (g -1⋅z )，（2）其中g -1=()abcd∈SU ()1, 2，g -1⋅z =(az +b )(cz +d )-1. 如果||ν>2且||ν∈Z +，那么πν是关于SU ()1, 2的离散序列，如果||ν>2，||ν∉Z +，则πν是万有覆盖群~SU ()1, 2的射影表示. 具体例子见文献（'Olafsson etal.，1988）.引理4（Zhang ，1992） 设L 2(B ，d μν(z ))d 是L 2(B ，d μν(z ))的闭子空间，且为离散序列表示，则L 2(B ，d μν(z ))d 有以下不可约分解L 2(B ，d μν(z ))d ≃∑||ν-22<r ≤||ν-2H 2r +4-||ν .177第 63 卷中山大学学报（自然科学版）（中英文）注3 当||ν≤2时，不存在离散序列表示；当||ν-2是正的偶整数时，有||ν-22-1个离散序列表示；当||ν-2是正的奇整数时，有||ν-12个离散序列表示.引理5（钟家庆，1989） 设l ≥0，a l =Γ(ρ+l )Γ(ρ)Γ(l +1)，且Re ρ>0，则(1-w )-ρ=∑l ≥0a lw l(w ∈C ， ||w <1).设()λ1，λ2为2l 的拆分，即(2l ，0)，(2l -1，1)，⋯，(l ，l )共有l +1种取法. 定义两个集合Λ+和Λ-满足ìíîλi ∈Λ+⇔λi -i 为奇数；λi ∈Λ-⇔λi -i 为偶数.为了行文方便，分别记Λ+以及Λ-中的元素为λi 1，λi 2，其中i 1，i 2∈{1，2}且i 1≠i 2.引理6（钟家庆，1989） 设l >0，z 1，z 2∈C ，则(z21+z 22)l=l ！∑λ1≥λ2≥0λ1+λ2=2lb ()λ1，λ2S ()λ1，λ2(z 1，z 2)，其中S (λ1，λ2)(z 1，z 2)=z λ1+11z λ22-z λ12z λ2+11z λ11z λ22-z λ12z λ21， b (λ1，λ2)=(-1)i 1-1()λi 1-i 1+12！()λi 2-i 2+22！，λi 1∈Λ+，λi 2∈Λ-.在引理6中，函数S (λ1，λ2)(z 1，z 2)称为Schur 函数，具体实例见文献（钟家庆，1989）. 定义Laplace-Beltrami 算子为L =∑j =12(z j∂∂z j )2+2∑i ≠j ()z 2i z i -z j∂∂z j .类似MacDonald （2015）的证明，对于2l =∑i =12λi ，有L S (λ1，λ2)(z 1，z 2)=C (λ1，λ2)S (λ1，λ2)(z 1，z 2)， C (λ1，λ2)=∑j =12λj (λj -2j )+3l ，且L 2S (λ1，λ2)(z 1，z 2)=C 2(λ1，λ2)S (λ1，λ2)(z 1，z 2).对于任意的z ∈B ，通过Laplace 方程的正则化理论，我们可定义H m (B )上的光滑闭子空间H ∞m (B )≔ìíîïïï∑l ≥0 ∑λ1≥λ2≥0λ1+λ2=la (λ1，λ2)S (λ1，λ2)(z 1，z 2) |||| ∑λ1≥λ2≥0||a(λ1，λ2)2C2(λ1，λ2)C (λ1，λ2)<+∞üýþïïïï，这里C (λ1，λ2)=∏j =12∏k =1λj(m -(j -1)+k -1).令m >2且m ∈2Z +，定义在B 上的分布向量为v 0(z )≔[]det (I 2-z tz )-m2.易证v 0(z )=(1-zzt)-m2=(1-z 21-z22)-m 2.命题2 设z ∈B ，m =2ρ. 记H -∞m (B )为H ∞m (B )的前对偶空间. 如果Re ρ>0，那么v 0(z )∈H -∞m (B ).证明 对于z ∈B ，利用引理2，引理3以及S (λ1，λ2)(z 1，z 2)的缓增系数，易得v 0(z )可以展开为收敛的幂级数形式. 从而在H ∞m (B )上赋予一些缓增系数得到其对应的连续线性泛函，有v 0(z )∈H -∞m (B )，命题得证.对于任意的g ∈SU ()1, 2和f ∈H -∞m (B )，定义逆步表示178第 1 期金四海，等：仿射对称空间SU(1，2)/SO(1，2)上的Plancherel 定理π∨m (g )f (⋅)=------------πm(g -1)f (⋅).（3）结合表示πν（见式（2）），可知v 0(z )是一个SO ()1, 2-不变的分布向量.命题3 设A p =exp ()a p ，对于任意的a t ∈A p ，有--------------------π∨m (a t )v 0(0)=1，π∨m (a t )v 0= cosh -m2(2t ).证明 设t ∈R ，且a t =()cosh ti sinh t 0-i sinh tcosh t 0001∈a p .利用式（2）~（3），可知--------------------π∨m (a t )v 0(0)=1，π∨m (a t )v 0=(cosh 2t +sinh 2t )-m 2=cosh -m2(2t ).引理7 对于任意g ∈SU ()1, 2，有Cartan-Berger 分解g =kah ，其中k ∈K ，a ∈A p ，h ∈SO ()1, 2.设x 0=e SO ()1, 2. 根据引理6、文献（'Olafsson et al.，1988），以及SU ()1，2/SO ()1，2=K exp (a +p )⋅x 0，有∫SU(1，2)/SO(1，2)f (x )d μ(x )=∫K ∫a+pf (kat⋅x 0)d μ0(t )d k ，其中d k 是K 上的规范Haar 测度，且d μ0(t )=4cosh (t )||sinh (t )cosh (2t )d t .（4）定理2 设m >2，则∫a +pcosh -m ()2t d μ0(t )=12Γ(1)Γ()m -22Γ()m 2.证明 利用式（4），有∫a +pcosh -m()2t d μ0(t )=∫∞2cosh (t )sinh (t )cosh1-m(2t )d (2t )=∫∞sinh ()2t cos h 1-m ()2t d ()2t .此外，对于Re(α)>-1且Re(β-α)>0，定义Beta 函数∫0∞sinh α(x )cosh -β(x )d x =12Γ()α+12Γ()β-α2Γ()β+12.取α=1，β=m -1，有∫a +pcosh -m ()2t d μ0(t )=12Γ(1)Γ()m -22Γ()m 2.利用Gamma 函数的性质，可知m >2，定理证毕.由引理3，定理2，可得以下结果：定理3 对于任意的h (z )∈H ∞2r +4-||ν(B )，则存在一个SO ()1, 2-不变的分布函数v 0∈H -∞2r +4-||ν(B )使得缠结算子I 2r +4-||ν(h )(g )=h ，π∨2r +4-||ν(g )v 0将H 2r +4-||ν(B )等距的延拓为F 2r +4-||ν，其中2r +4-||ν>2.证明 设m =2r +4-||ν. 对于ka t ∈KA p ，仅需找到Flensted-Jensen 函数1，π∨m ()ka t v 0，将其缠结化算子定义为I m 1(x )≔1，π∨m ()x v 0，179第 63 卷中山大学学报（自然科学版）（中英文）其中x ∈SU ()1，2/SO ()1，2. 根据文献（'Olafsson et al.，1988），容易得到空间F m ≔span{}|1，π∨m(ka t )v 0 ka t ∈KA p .此空间关于SU ()1，2的表示是不可约的. 于是，对于k ∈K ，a t ∈A p ，存在一个非零复数c 使得I m 1(ka t ⋅x 0)=cχ(k )cosh (t )-m 2，其中χ为K 的不可约特征标.现只需证明I m 1∈L 2()SU ()1，2/SO ()1，2. 为了证明这一事实，下文需要用到Bergmann 空间的一些结构，即循环向量的技术来表明此事实. 悉知，常值函数1在表示πm 意义下是H m (B )的循环向量，即对于任意的g ∈SU ()1，2，πm (g )1线性张成的空间在H m (B )中稠密. 对于1∈H m (B )，利用式（3）、引理4和定理3，可知m >2且m ∈2Z +，即F m 的指标与H m (B )的指标一致，可知F m ≃H m (B ). 因为特征标函数χ(k )是一个类函数，从而I m 1∈L 2()SU ()1，2/SO ()1，2，定理证毕.归结起来，由引理6以及定理3，定理1得证.参考文献：韩威， 范兴亚， 2023. 关于仿射对称空间SU （2，2）/SL （2，C ）+R 非紧分歧离散谱的一点注记［J ］. 新疆大学学报（自然科学版）（中英文）， 40（1）： 17-22+29.钟家庆， 1989. Schur 函数的一个展式及其在计数几何中的应用［J ］. 中国科学（A 辑）， （10）： 1018-1029.DELORME P ， KNOP F ， KRÖTZ B ， et al ， 2021. Plancherel theory for real spherical spaces ： Construction of the Bernstein mor ‐phisms ［J ］. J Amer Math Soc ， 34（3）： 815-908.DUISTERMAAT J J ， KOLK J A C ， VARADARAJAN V S ， 1979. Spectra of compact locally symmetric manifolds of negative cur ‐vature ［J ］. Invent Math ， 52（1）： 27-93.FLENSTED-JENSEN M ， 1980. Discrete series for semisimple symmetric spaces ［J ］. Ann Math ， 111（2）： 253-311.GEL'FAND I M ， GINDIKIN S G ， 1977. Complex manifolds whose skeletons are semisimple real Lie groups ， and analytic discreteseries of representations ［J ］. Funct Anal Its Appl ， 11（4）： 258-265.HARISH-CHANDRA ， 1965. Discrete series for semisimple Lie groups I ： Construction of invariant eigendistributions ［J ］. ActaMath ， 113： 241-318.KRÖTZ B ， KUIT J J ， OPDAM E M ， et al ， 2020. The infinitesimal characters of discrete series for real spherical spaces ［J ］. GeomFunct Anal ， 30（3）： 804-857.MACDONALD I G ， 2015. Symmetric functions and Hall polynomials ［M ］. 2th ed. New York ： Oxford University Press.'OLAFSSON G ， ØRSTED B ， 1988. The holomorphic discrete series for affine symmetric spaces ， I ［J ］. J Funct Anal ， 81（1）：126-159.ZHANG G ， 1992. A weighted Plancherel formula II ： The case of the ball ［J ］. Studia Math ， 102（2）： 103-120.（责任编辑 冯兆永）180。
Bi2Se3未考虑vdw的错误汇总
在没有考虑vdw作用之前，算Bi2Se3材料soc中出现的错误汇总V ASP自旋轨道耦合计算错误汇总静态计算时，报错：VERY BAD NEWS! Internal内部error in subroutine子程序IBZKPT:Reciprocal倒数的lattice and k-lattice belong to different class of lattices. Often results are still useful (48)INCAR参数设置：对策：根据所用集群，修改INCAR中NPAR。
将NPAR=4变成NPAR=1，已解决！错误：sub space matrix类错误报错：静态和能带计算中出现警告：W ARNING: Sub-Space-Matrix is not hermitian共轭in DA V结构优化出现错误：WARNING: Sub-Space-Matrix is not hermitian in DA V 4 -4.681828688433112E-002对策：通过将默认AMIX=0.4，修改成AMIX=0.2（或0.3），问题得以解决。
以下是类似的错误：WARNING: Sub-Space-Matrix is not hermitian in rmm -3.00000000000000RMM: 22 -0.167633596124E+02 -0.57393E+00 -0.44312E-01 1326 0.221E+00BRMIX:very serious problems the old and the new charge density differ old charge density: 28.00003 new 28.06093 0.111E+00错误：WARNING: Sub-Space-Matrix is not hermitian in rmm -42.5000000000000ERROR FEXCP: supplied Exchange-correletion table is too small, maximal index : 4794错误：结构优化Bi2Te3时，log文件：WARNING in EDDIAG: sub space matrix is not hermitian 1 -0.199E+01RMM: 200 0.179366581305E+01 -0.10588E-01 -0.14220E+00 718 0.261E-01BRMIX: very serious problems the old and the new charge density differ old charge density: 56.00230 new 124.70394 66 F= 0.17936658E+01 E0= 0.18295246E+01 d E =0.557217E-02curvature: 0.00 expect dE= 0.000E+00 dE for cont linesearch 0.000E+00ZBRENT: fatal error in bracketingplease rerun with smaller EDIFF, or copy CONTCAR to POSCAR and continue但是，将CONTCAR拷贝成POSCAR，接着算静态没有报错，这样算出来的结果有问题吗？对策1：用这个CONTCAR拷贝成POSCAR重新做一次结构优化，看是否达到优化精度！对策2：用这个CONTCAR拷贝成POSCAR，并且修改EDIFF（目前参数EDIFF=1E-6）,默认为10-4错误：WARNING: Sub-Space-Matrix is not hermitian in DA V 1 -7.626640664998020E-003网上参考解决方案：对策1：减小POTIM: IBRION=0，标准分子动力学模拟。
黑洞的准正模式(quasinormal modes)
Quasi-Normal Modes of Stars and Black HolesKostas D.KokkotasDepartment of Physics,Aristotle University of Thessaloniki,Thessaloniki54006,Greece.kokkotas@astro.auth.grhttp://www.astro.auth.gr/˜kokkotasandBernd G.SchmidtMax Planck Institute for Gravitational Physics,Albert Einstein Institute,D-14476Golm,Germany.bernd@aei-potsdam.mpg.dePublished16September1999/Articles/Volume2/1999-2kokkotasLiving Reviews in RelativityPublished by the Max Planck Institute for Gravitational PhysicsAlbert Einstein Institute,GermanyAbstractPerturbations of stars and black holes have been one of the main topics of relativistic astrophysics for the last few decades.They are of partic-ular importance today,because of their relevance to gravitational waveastronomy.In this review we present the theory of quasi-normal modes ofcompact objects from both the mathematical and astrophysical points ofview.The discussion includes perturbations of black holes(Schwarzschild,Reissner-Nordstr¨o m,Kerr and Kerr-Newman)and relativistic stars(non-rotating and slowly-rotating).The properties of the various families ofquasi-normal modes are described,and numerical techniques for calculat-ing quasi-normal modes reviewed.The successes,as well as the limits,of perturbation theory are presented,and its role in the emerging era ofnumerical relativity and supercomputers is discussed.c 1999Max-Planck-Gesellschaft and the authors.Further information on copyright is given at /Info/Copyright/.For permission to reproduce the article please contact livrev@aei-potsdam.mpg.de.Article AmendmentsOn author request a Living Reviews article can be amended to include errata and small additions to ensure that the most accurate and up-to-date infor-mation possible is provided.For detailed documentation of amendments, please go to the article’s online version at/Articles/Volume2/1999-2kokkotas/. Owing to the fact that a Living Reviews article can evolve over time,we recommend to cite the article as follows:Kokkotas,K.D.,and Schmidt,B.G.,“Quasi-Normal Modes of Stars and Black Holes”,Living Rev.Relativity,2,(1999),2.[Online Article]:cited on<date>, /Articles/Volume2/1999-2kokkotas/. The date in’cited on<date>’then uniquely identifies the version of the article you are referring to.3Quasi-Normal Modes of Stars and Black HolesContents1Introduction4 2Normal Modes–Quasi-Normal Modes–Resonances7 3Quasi-Normal Modes of Black Holes123.1Schwarzschild Black Holes (12)3.2Kerr Black Holes (17)3.3Stability and Completeness of Quasi-Normal Modes (20)4Quasi-Normal Modes of Relativistic Stars234.1Stellar Pulsations:The Theoretical Minimum (23)4.2Mode Analysis (26)4.2.1Families of Fluid Modes (26)4.2.2Families of Spacetime or w-Modes (30)4.3Stability (31)5Excitation and Detection of QNMs325.1Studies of Black Hole QNM Excitation (33)5.2Studies of Stellar QNM Excitation (34)5.3Detection of the QNM Ringing (37)5.4Parameter Estimation (39)6Numerical Techniques426.1Black Holes (42)6.1.1Evolving the Time Dependent Wave Equation (42)6.1.2Integration of the Time Independent Wave Equation (43)6.1.3WKB Methods (44)6.1.4The Method of Continued Fractions (44)6.2Relativistic Stars (45)7Where Are We Going?487.1Synergism Between Perturbation Theory and Numerical Relativity487.2Second Order Perturbations (48)7.3Mode Calculations (49)7.4The Detectors (49)8Acknowledgments50 9Appendix:Schr¨o dinger Equation Versus Wave Equation51Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt41IntroductionHelioseismology and asteroseismology are well known terms in classical astro-physics.From the beginning of the century the variability of Cepheids has been used for the accurate measurement of cosmic distances,while the variability of a number of stellar objects(RR Lyrae,Mira)has been associated with stel-lar oscillations.Observations of solar oscillations(with thousands of nonradial modes)have also revealed a wealth of information about the internal structure of the Sun[204].Practically every stellar object oscillates radially or nonradi-ally,and although there is great difficulty in observing such oscillations there are already results for various types of stars(O,B,...).All these types of pulsations of normal main sequence stars can be studied via Newtonian theory and they are of no importance for the forthcoming era of gravitational wave astronomy.The gravitational waves emitted by these stars are extremely weak and have very low frequencies(cf.for a discussion of the sun[70],and an im-portant new measurement of the sun’s quadrupole moment and its application in the measurement of the anomalous precession of Mercury’s perihelion[163]). This is not the case when we consider very compact stellar objects i.e.neutron stars and black holes.Their oscillations,produced mainly during the formation phase,can be strong enough to be detected by the gravitational wave detectors (LIGO,VIRGO,GEO600,SPHERE)which are under construction.In the framework of general relativity(GR)quasi-normal modes(QNM) arise,as perturbations(electromagnetic or gravitational)of stellar or black hole spacetimes.Due to the emission of gravitational waves there are no normal mode oscillations but instead the frequencies become“quasi-normal”(complex), with the real part representing the actual frequency of the oscillation and the imaginary part representing the damping.In this review we shall discuss the oscillations of neutron stars and black holes.The natural way to study these oscillations is by considering the linearized Einstein equations.Nevertheless,there has been recent work on nonlinear black hole perturbations[101,102,103,104,100]while,as yet nothing is known for nonlinear stellar oscillations in general relativity.The study of black hole perturbations was initiated by the pioneering work of Regge and Wheeler[173]in the late50s and was continued by Zerilli[212]. The perturbations of relativistic stars in GR werefirst studied in the late60s by Kip Thorne and his collaborators[202,198,199,200].The initial aim of Regge and Wheeler was to study the stability of a black hole to small perturbations and they did not try to connect these perturbations to astrophysics.In con-trast,for the case of relativistic stars,Thorne’s aim was to extend the known properties of Newtonian oscillation theory to general relativity,and to estimate the frequencies and the energy radiated as gravitational waves.QNMs werefirst pointed out by Vishveshwara[207]in calculations of the scattering of gravitational waves by a Schwarzschild black hole,while Press[164] coined the term quasi-normal frequencies.QNM oscillations have been found in perturbation calculations of particles falling into Schwarzschild[73]and Kerr black holes[76,80]and in the collapse of a star to form a black hole[66,67,68]. Living Reviews in Relativity(1999-2)5Quasi-Normal Modes of Stars and Black Holes Numerical investigations of the fully nonlinear equations of general relativity have provided results which agree with the results of perturbation calculations;in particular numerical studies of the head-on collision of two black holes [30,29](cf.Figure 1)and gravitational collapse to a Kerr hole [191].Recently,Price,Pullin and collaborators [170,31,101,28]have pushed forward the agreement between full nonlinear numerical results and results from perturbation theory for the collision of two black holes.This proves the power of the perturbation approach even in highly nonlinear problems while at the same time indicating its limits.In the concluding remarks of their pioneering paper on nonradial oscillations of neutron stars Thorne and Campollataro [202]described it as “just a modest introduction to a story which promises to be long,complicated and fascinating ”.The story has undoubtedly proved to be intriguing,and many authors have contributed to our present understanding of the pulsations of both black holes and neutron stars.Thirty years after these prophetic words by Thorne and Campollataro hundreds of papers have been written in an attempt to understand the stability,the characteristic frequencies and the mechanisms of excitation of these oscillations.Their relevance to the emission of gravitational waves was always the basic underlying reason of each study.An account of all this work will be attempted in the next sections hoping that the interested reader will find this review useful both as a guide to the literature and as an inspiration for future work on the open problems of the field.020406080100Time (M ADM )-0.3-0.2-0.10.00.10.20.3(l =2) Z e r i l l i F u n c t i o n Numerical solutionQNM fit Figure 1:QNM ringing after the head-on collision of two unequal mass black holes [29].The continuous line corresponds to the full nonlinear numerical calculation while the dotted line is a fit to the fundamental and first overtone QNM.In the next section we attempt to give a mathematical definition of QNMs.Living Reviews in Relativity (1999-2)K.D.Kokkotas and B.G.Schmidt6 The third and fourth section will be devoted to the study of the black hole and stellar QNMs.In thefifth section we discuss the excitation and observation of QNMs andfinally in the sixth section we will mention the more significant numerical techniques used in the study of QNMs.Living Reviews in Relativity(1999-2)7Quasi-Normal Modes of Stars and Black Holes 2Normal Modes–Quasi-Normal Modes–Res-onancesBefore discussing quasi-normal modes it is useful to remember what normal modes are!Compact classical linear oscillating systems such asfinite strings,mem-branes,or cavitiesfilled with electromagnetic radiation have preferred time harmonic states of motion(ωis real):χn(t,x)=e iωn tχn(x),n=1,2,3...,(1) if dissipation is neglected.(We assumeχto be some complex valuedfield.) There is generally an infinite collection of such periodic solutions,and the“gen-eral solution”can be expressed as a superposition,χ(t,x)=∞n=1a n e iωn tχn(x),(2)of such normal modes.The simplest example is a string of length L which isfixed at its ends.All such systems can be described by systems of partial differential equations of the type(χmay be a vector)∂χ∂t=Aχ,(3)where A is a linear operator acting only on the spatial variables.Because of thefiniteness of the system the time evolution is only determined if some boundary conditions are prescribed.The search for solutions periodic in time leads to a boundary value problem in the spatial variables.In simple cases it is of the Sturm-Liouville type.The treatment of such boundary value problems for differential equations played an important role in the development of Hilbert space techniques.A Hilbert space is chosen such that the differential operator becomes sym-metric.Due to the boundary conditions dictated by the physical problem,A becomes a self-adjoint operator on the appropriate Hilbert space and has a pure point spectrum.The eigenfunctions and eigenvalues determine the periodic solutions(1).The definition of self-adjointness is rather subtle from a physicist’s point of view since fairly complicated“domain issues”play an essential role.(See[43] where a mathematical exposition for physicists is given.)The wave equation modeling thefinite string has solutions of various degrees of differentiability. To describe all“realistic situations”,clearly C∞functions should be sufficient. Sometimes it may,however,also be convenient to consider more general solu-tions.From the mathematical point of view the collection of all smooth functions is not a natural setting to study the wave equation because sequences of solutionsLiving Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt8 exist which converge to non-smooth solutions.To establish such powerful state-ments like(2)one has to study the equation on certain subsets of the Hilbert space of square integrable functions.For“nice”equations it usually happens that the eigenfunctions are in fact analytic.They can then be used to gen-erate,for example,all smooth solutions by a pointwise converging series(2). The key point is that we need some mathematical sophistication to obtain the “completeness property”of the eigenfunctions.This picture of“normal modes”changes when we consider“open systems”which can lose energy to infinity.The simplest case are waves on an infinite string.The general solution of this problem isχ(t,x)=A(t−x)+B(t+x)(4) with“arbitrary”functions A and B.Which solutions should we study?Since we have all solutions,this is not a serious question.In more general cases, however,in which the general solution is not known,we have to select a certain class of solutions which we consider as relevant for the physical problem.Let us consider for the following discussion,as an example,a wave equation with a potential on the real line,∂2∂t2χ+ −∂2∂x2+V(x)χ=0.(5)Cauchy dataχ(0,x),∂tχ(0,x)which have two derivatives determine a unique twice differentiable solution.No boundary condition is needed at infinity to determine the time evolution of the data!This can be established by fairly simple PDE theory[116].There exist solutions for which the support of thefields are spatially compact, or–the other extreme–solutions with infinite total energy for which thefields grow at spatial infinity in a quite arbitrary way!From the point of view of physics smooth solutions with spatially compact support should be the relevant class–who cares what happens near infinity! Again it turns out that mathematically it is more convenient to study all solu-tions offinite total energy.Then the relevant operator is again self-adjoint,but now its spectrum is purely“continuous”.There are no eigenfunctions which are square integrable.Only“improper eigenfunctions”like plane waves exist.This expresses the fact that wefind a solution of the form(1)for any realωand by forming appropriate superpositions one can construct solutions which are “almost eigenfunctions”.(In the case V(x)≡0these are wave packets formed from plane waves.)These solutions are the analogs of normal modes for infinite systems.Let us now turn to the discussion of“quasi-normal modes”which are concep-tually different to normal modes.To define quasi-normal modes let us consider the wave equation(5)for potentials with V≥0which vanish for|x|>x0.Then in this case all solutions determined by data of compact support are bounded: |χ(t,x)|<C.We can use Laplace transformation techniques to represent such Living Reviews in Relativity(1999-2)9Quasi-Normal Modes of Stars and Black Holes solutions.The Laplace transformˆχ(s,x)(s>0real)of a solutionχ(t,x)isˆχ(s,x)= ∞0e−stχ(t,x)dt,(6) and satisfies the ordinary differential equations2ˆχ−ˆχ +Vˆχ=+sχ(0,x)+∂tχ(0,x),(7) wheres2ˆχ−ˆχ +Vˆχ=0(8) is the homogeneous equation.The boundedness ofχimplies thatˆχis analytic for positive,real s,and has an analytic continuation onto the complex half plane Re(s)>0.Which solutionˆχof this inhomogeneous equation gives the unique solution in spacetime determined by the data?There is no arbitrariness;only one of the Green functions for the inhomogeneous equation is correct!All Green functions can be constructed by the following well known method. Choose any two linearly independent solutions of the homogeneous equation f−(s,x)and f+(s,x),and defineG(s,x,x )=1W(s)f−(s,x )f+(s,x)(x <x),f−(s,x)f+(s,x )(x >x),(9)where W(s)is the Wronskian of f−and f+.If we denote the inhomogeneity of(7)by j,a solution of(7)isˆχ(s,x)= ∞−∞G(s,x,x )j(s,x )dx .(10) We still have to select a unique pair of solutions f−,f+.Here the information that the solution in spacetime is bounded can be used.The definition of the Laplace transform implies thatˆχis bounded as a function of x.Because the potential V vanishes for|x|>x0,the solutions of the homogeneous equation(8) for|x|>x0aref=e±sx.(11) The following pair of solutionsf+=e−sx for x>x0,f−=e+sx for x<−x0,(12) which is linearly independent for Re(s)>0,gives the unique Green function which defines a bounded solution for j of compact support.Note that for Re(s)>0the solution f+is exponentially decaying for large x and f−is expo-nentially decaying for small x.For small x however,f+will be a linear com-bination a(s)e−sx+b(s)e sx which will in general grow exponentially.Similar behavior is found for f−.Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt 10Quasi-Normal mode frequencies s n can be defined as those complex numbers for whichf +(s n ,x )=c (s n )f −(s n ,x ),(13)that is the two functions become linearly dependent,the Wronskian vanishes and the Green function is singular!The corresponding solutions f +(s n ,x )are called quasi eigenfunctions.Are there such numbers s n ?From the boundedness of the solution in space-time we know that the unique Green function must exist for Re (s )>0.Hence f +,f −are linearly independent for those values of s .However,as solutions f +,f −of the homogeneous equation (8)they have a unique continuation to the complex s plane.In [35]it is shown that for positive potentials with compact support there is always a countable number of zeros of the Wronskian with Re (s )<0.What is the mathematical and physical significance of the quasi-normal fre-quencies s n and the corresponding quasi-normal functions f +?First of all we should note that because of Re (s )<0the function f +grows exponentially for small and large x !The corresponding spacetime solution e s n t f +(s n ,x )is therefore not a physically relevant solution,unlike the normal modes.If one studies the inverse Laplace transformation and expresses χas a com-plex line integral (a >0),χ(t,x )=12πi +∞−∞e (a +is )t ˆχ(a +is,x )ds,(14)one can deform the path of the complex integration and show that the late time behavior of solutions can be approximated in finite parts of the space by a finite sum of the form χ(t,x )∼N n =1a n e (αn +iβn )t f +(s n ,x ).(15)Here we assume that Re (s n +1)<Re (s n )<0,s n =αn +iβn .The approxi-mation ∼means that if we choose x 0,x 1, and t 0then there exists a constant C (t 0,x 0,x 1, )such that χ(t,x )−N n =1a n e (αn +iβn )t f +(s n ,x ) ≤Ce (−|αN +1|+ )t (16)holds for t >t 0,x 0<x <x 1, >0with C (t 0,x 0,x 1, )independent of t .The constants a n depend only on the data [35]!This implies in particular that all solutions defined by data of compact support decay exponentially in time on spatially bounded regions.The generic leading order decay is determined by the quasi-normal mode frequency with the largest real part s 1,i.e.slowest damping.On finite intervals and for late times the solution is approximated by a finite sum of quasi eigenfunctions (15).It is presently unclear whether one can strengthen (16)to a statement like (2),a pointwise expansion of the late time solution in terms of quasi-normal Living Reviews in Relativity (1999-2)11Quasi-Normal Modes of Stars and Black Holes modes.For one particular potential(P¨o schl-Teller)this has been shown by Beyer[42].Let us now consider the case where the potential is positive for all x,but decays near infinity as happens for example for the wave equation on the static Schwarzschild spacetime.Data of compact support determine again solutions which are bounded[117].Hence we can proceed as before.Thefirst new point concerns the definitions of f±.It can be shown that the homogeneous equation(8)has for each real positive s a unique solution f+(s,x)such that lim x→∞(e sx f+(s,x))=1holds and correspondingly for f−.These functions are uniquely determined,define the correct Green function and have analytic continuations onto the complex half plane Re(s)>0.It is however quite complicated to get a good representation of these func-tions.If the point at infinity is not a regular singular point,we do not even get converging series expansions for f±.(This is particularly serious for values of s with negative real part because we expect exponential growth in x).The next new feature is that the analyticity properties of f±in the complex s plane depend on the decay of the potential.To obtain information about analytic continuation,even use of analyticity properties of the potential in x is made!Branch cuts may occur.Nevertheless in a lot of cases an infinite number of quasi-normal mode frequencies exists.The fact that the potential never vanishes may,however,destroy the expo-nential decay in time of the solutions and therefore the essential properties of the quasi-normal modes.This probably happens if the potential decays slower than exponentially.There is,however,the following way out:Suppose you want to study a solution determined by data of compact support from t=0to some largefinite time t=T.Up to this time the solution is–because of domain of dependence properties–completely independent of the potential for sufficiently large x.Hence we may see an exponential decay of the form(15)in a time range t1<t<T.This is the behavior seen in numerical calculations.The situation is similar in the case ofα-decay in quantum mechanics.A comparison of quasi-normal modes of wave equations and resonances in quantum theory can be found in the appendix,see section9.Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt123Quasi-Normal Modes of Black HolesOne of the most interesting aspects of gravitational wave detection will be the connection with the existence of black holes[201].Although there are presently several indirect ways of identifying black holes in the universe,gravitational waves emitted by an oscillating black hole will carry a uniquefingerprint which would lead to the direct identification of their existence.As we mentioned earlier,gravitational radiation from black hole oscillations exhibits certain characteristic frequencies which are independent of the pro-cesses giving rise to these oscillations.These“quasi-normal”frequencies are directly connected to the parameters of the black hole(mass,charge and angu-lar momentum)and for stellar mass black holes are expected to be inside the bandwidth of the constructed gravitational wave detectors.The perturbations of a Schwarzschild black hole reduce to a simple wave equation which has been studied extensively.The wave equation for the case of a Reissner-Nordstr¨o m black hole is more or less similar to the Schwarzschild case,but for Kerr one has to solve a system of coupled wave equations(one for the radial part and one for the angular part).For this reason the Kerr case has been studied less thoroughly.Finally,in the case of Kerr-Newman black holes we face the problem that the perturbations cannot be separated in their angular and radial parts and thus apart from special cases[124]the problem has not been studied at all.3.1Schwarzschild Black HolesThe study of perturbations of Schwarzschild black holes assumes a small per-turbation hµνon a static spherically symmetric background metricds2=g0µνdxµdxν=−e v(r)dt2+eλ(r)dr2+r2 dθ2+sin2θdφ2 ,(17) with the perturbed metric having the formgµν=g0µν+hµν,(18) which leads to a variation of the Einstein equations i.e.δGµν=4πδTµν.(19) By assuming a decomposition into tensor spherical harmonics for each hµνof the formχ(t,r,θ,φ)= mχ m(r,t)r Y m(θ,φ),(20)the perturbation problem is reduced to a single wave equation,for the func-tionχ m(r,t)(which is a combination of the various components of hµν).It should be pointed out that equation(20)is an expansion for scalar quantities only.From the10independent components of the hµνonly h tt,h tr,and h rr transform as scalars under rotations.The h tθ,h tφ,h rθ,and h rφtransform asLiving Reviews in Relativity(1999-2)13Quasi-Normal Modes of Stars and Black Holes components of two-vectors under rotations and can be expanded in a series of vector spherical harmonics while the components hθθ,hθφ,and hφφtransform as components of a2×2tensor and can be expanded in a series of tensor spher-ical harmonics(see[202,212,152]for details).There are two classes of vector spherical harmonics(polar and axial)which are build out of combinations of the Levi-Civita volume form and the gradient operator acting on the scalar spherical harmonics.The difference between the two families is their parity. Under the parity operatorπa spherical harmonic with index transforms as (−1) ,the polar class of perturbations transform under parity in the same way, as(−1) ,and the axial perturbations as(−1) +11.Finally,since we are dealing with spherically symmetric spacetimes the solution will be independent of m, thus this subscript can be omitted.The radial component of a perturbation outside the event horizon satisfies the following wave equation,∂2∂t χ + −∂2∂r∗+V (r)χ =0,(21)where r∗is the“tortoise”radial coordinate defined byr∗=r+2M log(r/2M−1),(22) and M is the mass of the black hole.For“axial”perturbationsV (r)= 1−2M r ( +1)r+2σMr(23)is the effective potential or(as it is known in the literature)Regge-Wheeler potential[173],which is a single potential barrier with a peak around r=3M, which is the location of the unstable photon orbit.The form(23)is true even if we consider scalar or electromagnetic testfields as perturbations.The parameter σtakes the values1for scalar perturbations,0for electromagnetic perturbations, and−3for gravitational perturbations and can be expressed asσ=1−s2,where s=0,1,2is the spin of the perturbingfield.For“polar”perturbations the effective potential was derived by Zerilli[212]and has the form V (r)= 1−2M r 2n2(n+1)r3+6n2Mr2+18nM2r+18M3r3(nr+3M)2,(24)1In the literature the polar perturbations are also called even-parity because they are characterized by their behavior under parity operations as discussed earlier,and in the same way the axial perturbations are called odd-parity.We will stick to the polar/axial terminology since there is a confusion with the definition of the parity operation,the reason is that to most people,the words“even”and“odd”imply that a mode transforms underπas(−1)2n or(−1)2n+1respectively(for n some integer).However only the polar modes with even have even parity and only axial modes with even have odd parity.If is odd,then polar modes have odd parity and axial modes have even parity.Another terminology is to call the polar perturbations spheroidal and the axial ones toroidal.This definition is coming from the study of stellar pulsations in Newtonian theory and represents the type offluid motions that each type of perturbation induces.Since we are dealing both with stars and black holes we will stick to the polar/axial terminology.Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt14where2n=( −1)( +2).(25) Chandrasekhar[54]has shown that one can transform the equation(21)for “axial”modes to the corresponding one for“polar”modes via a transforma-tion involving differential operations.It can also be shown that both forms are connected to the Bardeen-Press[38]perturbation equation derived via the Newman-Penrose formalism.The potential V (r∗)decays exponentially near the horizon,r∗→−∞,and as r−2∗for r∗→+∞.From the form of equation(21)it is evident that the study of black hole perturbations will follow the footsteps of the theory outlined in section2.Kay and Wald[117]have shown that solutions with data of compact sup-port are bounded.Hence we know that the time independent Green function G(s,r∗,r ∗)is analytic for Re(s)>0.The essential difficulty is now to obtain the solutions f±(cf.equation(10))of the equations2ˆχ−ˆχ +Vˆχ=0,(26) (prime denotes differentiation with respect to r∗)which satisfy for real,positives:f+∼e−sr∗for r∗→∞,f−∼e+r∗x for r∗→−∞.(27) To determine the quasi-normal modes we need the analytic continuations of these functions.As the horizon(r∗→∞)is a regular singular point of(26),a representation of f−(r∗,s)as a converging series exists.For M=12it reads:f−(r,s)=(r−1)s∞n=0a n(s)(r−1)n.(28)The series converges for all complex s and|r−1|<1[162].(The analytic extension of f−is investigated in[115].)The result is that f−has an extension to the complex s plane with poles only at negative real integers.The representation of f+is more complicated:Because infinity is a singular point no power series expansion like(28)exists.A representation coming from the iteration of the defining integral equation is given by Jensen and Candelas[115],see also[159]. It turns out that the continuation of f+has a branch cut Re(s)≤0due to the decay r−2for large r[115].The most extensive mathematical investigation of quasi-normal modes of the Schwarzschild solution is contained in the paper by Bachelot and Motet-Bachelot[35].Here the existence of an infinite number of quasi-normal modes is demonstrated.Truncating the potential(23)to make it of compact support leads to the estimate(16).The decay of solutions in time is not exponential because of the weak decay of the potential for large r.At late times,the quasi-normal oscillations are swamped by the radiative tail[166,167].This tail radiation is of interest in its Living Reviews in Relativity(1999-2)。
百牛定理HecatombProposition
西方的名稱
但是把 m, n 加以限制為兩個「互素」(即「互質」, co-prime) 的奇數，就可以造
商高定理 陳子定理 勾股定理
畢氏定理﹙畢達哥拉斯定理﹚ 百牛定理 木匠定理
出全部兩兩互素的「勾股數組」。 仔細觀察「勾股數組」，它們總是具有一定的奇偶關係，也就是二奇一偶。事
實上，如果 a, b, c 是一組兩兩互素的勾股數，那麼 a, b 必定一奇一偶，c 必為 奇數。
-1 + 2
5 ，而近似值為 0.618。這就是黃金比例了。
在人體軀幹與身高的比例上，肚臍是理想的黃金分割點。換言之，若此比值愈 接近 0.618，愈給與人一種美的感覺。很可惜，一般人的軀幹(由腳底至肚臍的長 度)與身高比都低於此數值，大約只有 0.518 至 0.60 左右(腳長的人會有較高的比 值)。所以有很多人要穿高跟鞋。
摘自【十萬個為甚麼⎯⎯數學篇 I，新世紀版】
仁愛堂田家炳中學 中二級 數學科 第十章 畢氏定理
黃金比例
在自然界裏，物體形狀的比例提供了在均稱和協調上一種美感的參考。在數 學上，這個比例稱為黃金分割。
在線段 AB 上，若要找出黃金分割的位置，可以設分割點 G，G 會符合以下
的特性：AB : AG = AG : GB
B
設 AB = l ; AG = x
G
則 l : x = x : (l – x)
数学-科学的王后与仆人
数学: 科学的王后和仆人Mathematics: Queen and Servant of Science北京理工大学叶其孝本文的题目是已故的美国科学院院士、著名数学家、数学史学家和科普作家Eric Temple Bell(贝尔, 1883, 02, 07 ~ 1960, 12, 21)于1951年写的一本书的书名Mathematics: Queen and Servant of Science (数学: 科学的王后和仆人). 该书主要是为大学生和非数学领域的人士写的, 介绍纯粹和应用数学的各个方面, 更着重在说明数学科学的极端重要性.The Mathematical Association of America, 1996, 463 pages实际上这是他1931年写的The Queen of the Sciences (科学的王后)和1937年写的The Handmaiden of the Sciences (科学的女仆)这两本通俗数学论著的合一修订扩大版.Eric Temple Bell Alexander Graham Bell (1847 ~ 1922) 按常识的理解, 女王是优美、高雅、无懈可击、至尊至贵的, 在科学中只有纯粹数学才具有这样的特点, 简洁明了的数学定理一经证明就是永恒的真理, 极其优美而且无懈可击;另一方面, 科学和工程的各个分支都在不同程度上大量应用数学, 这时数学科学就是仆人, 这些仆人是否强有力, 用起来是否得心应手是雇佣这些仆人的主人最为关心的事. 事实上, servant这个字本身就有“供人们利用之物, 有用的服务工具”的意思. 毫无疑问, 我们的目的不是为数学争一个好的名分, 而是想说明数学是怎样通过数学建模来解决各种实际问题的; 数学(数学建模)的极端重要性, 以及探讨正确认识和理解数学科学的作用对于发展我国科学技术、经济以及教育, 从而争取在21世纪把我国真正建设成为屹立于世界民族之林的强国，乃至个人事业发展的至关重要性. 当然, 我们也希望说明王后和仆人集于一身并不矛盾. 历史上, 很多特别受人尊敬的科学家, 不仅仅是由于他们的科学成就, 更因为他们的科学成就能够服务于人类.数学是科学的王后, 算术是数学的王后. 她常常放下架子为天文学和其他科学效劳, 但是在所有情况下, 第一位的是她(数学)应尽的责任. (高斯)Mathematics is the Queen of the Sciences, and Arithmetic the Queen of Mathematics. She often condescends to render service to astronomy and other natural sciences, but under all circumstance the first place is her due.— Carl Friedrich Gauss (卡尔·弗里德里希·高斯, 1777, 4, 30 ~ 1855, 2, 23)From: Bell, Eric T., Mathematics: Queen and Servant of Science, MAA, 1951, p.1;Men of Mathematics, Simon and Schuster, New York, 1937, p. xv.***************************************************自古以来，数学的发展始终与科学技术的发展紧密相连，反之亦然. 首先, 我们来看一下导致我们现在这个飞速发展的信息社会的19、20世纪几乎所有重大科学理论的发展和完善过程中数学(数学建模)所起到的不可勿缺的作用.数学研究的成果往往是重大科学发明的催生素(仅就19、20世纪而言, 流体力学、电磁理论、相对论、量子力学、计算机、信息论、控制论、现代经济学、万维网和互联网搜索引擎、生物学、CT、甚至社会政治学领域等). 但是20世纪上半世纪, 数学虽然也直接为工程技术提供一些工具, 但基本方式是间接的: 先促进其他科学的发展, 再由这些科学提供工程原理和设计的基础. 数学是幕后的无名英雄.现在, 数学无处不在, 数学和工程技术之间,在更广阔的范围内和更深刻的程度上, 直接地相互作用着, 极大地推动了科学和工程科学的发展, 也极大地推动了技术的发展. 数学不仅是幕后的无名英雄, 很多方面开始走向“前台”. 但是对数学的极端重要性迄今尚未有共识, 取得共识对加强一个国家的竞争力来说是至关重要的.硬能力―一位美国朋友谈及对未来中国人的看法: 20年后, 中国年轻人会丢了中国人现在的硬能力, 他们崇拜各种明星, 不愿献身科学, 不再以学术研究为荣, 聪明拔尖的学生都去学金融、法律等赚钱的专业; 而美国人因为认识到其硬能力(例如数学)不行, 进行教育改革, 20年后, 不但保持了其软实力即非专业能力的优势, 而且在硬能力上赶上中国人.‖“正在丢失的硬实力”, 鲁鸣, 《青年文摘》2011年第5期动向：美国很多州新办STEM高中, 一些大学开始开设STEM课程等.STEM = Science + Technology + Engineering + Mathematics2012年2月7日公布的美国总统科技顾问委员会给总统的报告，参与超越：培养额外的100万具有科学、技术、工程和数学学位的大学生(Engage to Excel: Producing One Million Additional College Graduates with Degrees in Science, Technology, Engineering, and Mathematics)The Mathematical Sciences in 2025, the National Academies Press, 2013人们使用的数学科学思想、概念和方法的范围在不断扩大的同时，数学科学的用途也在不断扩展. 21世纪的大部分科学与工程将建立在数学科学的基础上.This major expansion in the uses of the mathematical sciences has been paralleled by a broadening in the range of mathematical science ideas and techniques being used. Much of twenty-first century science and engineering is going to be built on a mathematical science foundation, and that foundation must continue to evolve and expand.数学科学是日常生活的几乎每个方面的组成部分.互联网搜索、医疗成像、电脑动画、数值天气预报和其他计算机模拟、所有类型的数字通信、商业和军事中的优化问题以及金融风险的分析——普通公民都从支撑这些应用功能的数学科学的各种进展中获益，这样的例子不胜枚举.The mathematical sciences are part of almost every aspect of everyday life. Internet search, medical imaging, computer animation, numerical weather predictions and othercomputer simulations, digital communications of all types, optimization in business and the military, analyses of financial risks —average citizens all benefit from the mathematical science advances that underpin these capabilities, and the list goes on and on.调查发现:数学科学研究工作正日益成为生物学、医学、社会科学、商业、先进设计、气候、金融、先进材料等许多研究领域不可或缺的重要组成部分. 这种研究工作涉及最广泛意义下数学、统计学和计算综合，以及这些领域与潜在应用领域的相互作用. 所有这些活动对于经济增长、国家竞争力和国家安全都是至关重要的，而且这种事实应该对作为整体的数学科学的资助性质和资助规模产生影响. 数学科学的教育也应该反映数学科学领域的新的状况.Finding: Mathematical sciences work is becoming an increasingly integral and essential component of a growing array of areas of investigation in biology, medicine, social sciences, business, advanced design, climate, finance, advanced materials, and many more. This work involves the integration of mathematics, statistics, and computation in the broadest sense and the interplay of these areas withareas of potential application. All of these activities are crucial to economic growth, national competitiveness, and national security, and this fact should inform both the nature and scale of funding for the mathematical sciences as a whole. Education in the mathematical sciences should also reflect this new stature of the field.****************************************************************为了以下讲述的方便, 我们先来了解一下什么是数学建模.数学模型(Mathematical Model)是用数学符号对一类实际问题或实际发生的现象的(近似的)描述.数学建模(Mathematical Modeling)则是获得该模型并对之求解、验证并得到结论的全过程.数学建模不仅是了解基本规律, 而且从应用的观点来看更重要的是预测和控制所建模的系统的行为的强有力的工具.数学建模是数学用来解决各种实际问题的桥梁.↑→→→→→→→→↓↑↓↑↓↓↑↓←←←←←通不过↓↓通过)定义：数学建模就是上述框图多次执行的过程数学建模的难点观察、分析实际问题, 作出合理的假设, 明确变量和参数, 形成明确的数学问题. 不仅仅是翻译的问题; 涉及的数学问题可能是复杂、困难的, 求解也许涉及深刻的数学方法. 如何作出正确的判断, 寻找合适、简洁的(解析或近似) 解法; 如何验证模型.简言之:合理假设、模型建立、模型求解、解释验证.记住这16个字, 将会终生受用.数学建模的重要作用：源头创新当然数学建模也有局限性, 不能单独包打天下, 因为实际问题是非常复杂的, 需要多学科协同解决.在图灵(A. M. Turing)的文章: The Chemical Basis of Morphogenesis (形态生成的化学基础), Philosophical Transactions of the Royal Society of London (伦敦皇家学会哲学公报), Series B (Biological Sciences),v.237(1952), 37-72.1. 一个胚胎的模型. 成形素本节将描述一个正在生长的胚胎的数学模型. 该模型是一种简化和理想化, 因此是对原问题的篡改. 希望本文论述中保留的一些特征, 就现今的知识状况而言, 是那些最重要的特征.1. A model of the embryo. MorphogensIn this section a mathematical model of the growing embryo will be described. This model will be asimplification and an idealization, and consequently a falsification. It is to be hoped that the features retained for discussion are those of greatest importance in the present state of knowledge.想单靠数学建模本身来解决重大的生物学问题是不可能的，另一方面，想仅仅依靠实验来获得对生物学的合理、完整的理解也是极不可能的. There is no way mathematical modeling can solve major biological problems on its own. On the other hand, it ishighly unlikely that even a reasonably complete understanding could come solely from experiment.—— J. D. Murray, Why Are There No 3-Headed Monsters? Mathematical Modeling in Biology, Notices of the AMS,v. 59 (2012), no. 6, p.793.自古以来公平、公正的竞赛都是培养、选拔人才的重要手段, 科学和数学也不例外.中学生IMO (国际数学奥林匹克(International Mathematical Olympiad), 1959 ～)北美的大学生Putnbam数学竞赛(1938 ～)全国大学生数学竞赛(2010 ～)Mathematical Contest in Modeling (MCM, 1985 ～)美国大学生数学建模竞赛Interdisciplinary Contest in Modeling (ICM, 1999～)美国大学生跨学科建模竞赛China Undergraduate Mathematical Contest in Modeling (CUMCM, 1992～) 中国大学生数学建模竞赛中国大学生参加美国大学生数学建模竞赛情况中国大学生数学建模竞赛情况在以下讲述中涉及物理方面的具体的数学模型 (问题)的叙述和初步讨论可参考《物理学与偏微分方程》, 李大潜、秦铁虎编著, (上册, 1997; 下册, 2000), 高等教育出版社.Seven equations that rule your world (主宰你生活的七个方程式), by Ian Stewart, NewScientist, 13 February 2012.Fourier transformation 2ˆ()()ix f f x e dx πξξ∞--∞=⎰Wave equation 22222u u c t x ∂∂=∂∂ Ma xwell‘s equation110, , 0, H E E E H H c t c t∂∂∇⋅=∇⨯=-∇⋅=∇⨯=∂∂Schrödinger‘s equation ˆψH ψi t∂=∂Ian Stewart, In Pursuit of the Unknown:17 Equations That Changed the World (追求对未知的认识：改变世界的17个方程), Basic Books, March 13, 2012.目录(Contents)Why Equations? /viii1. The squaw on the hippopotamus ——Pythagoras‘sTheorem/12. Shortening the proceedings —— Logarithms/213. Ghosts of departed quantities —— Calculus/354. The system of the world ——Newton‘s Law ofGravity/535. Portent of the ideal world —— The Square Root ofMinus One/736. Much ado about knotting ——Euler‘s Formula forPolyhedra/837. Patterns of chance —— Normal Distribution/1078. Good vibrations —— Wave Equation/1319. Ripples and blips —— Fourier Transform/14910. The ascent of humanity —— Navier-StokesEquation/16511. Wave in the ether ——Maxwell‘s Equations/17912. Law and disorder —— Second Law ofThermodynamics /19513. One thing is absolute —— Relativity/21714. Quantum weirdness —— Schrödinger Equation/24515. Codes, communications, and computers ——Information Theory/26516. The imbalance of nature —— Chaos Theory/28317. The Midas formula —— Black-Scholes Equation/195Where Next?/317Notes/321Illustration Credits/330Index/331相对论Albert Einstein(1879, 3, 14 ～1955, 4, 18)20世纪最伟大的科学成就莫过于Einstein(爱因斯坦)的狭义和广义相对论了, 但是如果没有Minkowski (闵可夫斯基)几何、Riemann(黎曼)于1854年发明的Riemann几何, 以及Cayley(凯莱), Sylvester(西勒维斯特)和Noether(诺特)等数学家发展的不变量理论, Einstein的广义相对论和引力理论就不可能有如此完善的数学表述. Einstein自己也不止一次地说过.早在1905年, 年仅26岁的爱因斯坦就已提出了狭义相对论. 狭义相对论推倒了牛顿力学的质量守恒、能量守恒、质量能量互不相关、时空永恒不变的基本命题. 这是一场真正的科学革命.为了导出狭义相对论，爱因斯坦作出了两个假设：运动的相对性(所有匀速运动都是相对的)和光速为常数(光的运动例外, 它是绝对的). (1)狭义相对性原理,即在所有惯性系中, 物理学定律具有相同的数学表达形式;(2)光速不变原理,真空中光沿各个方向传播的速率都相等，与光源和观察者的运动状态无关.时空不是绝对独立的.由此可以导出一些推论: 相对论坐标变换式和速度变换式, 同时的相对性, 钟慢尺缩效应和质能关系式等.他的好友物理学家P.Ehrenfest指出实际上还蕴涵着第三个假设, 即这两个假设是不矛盾的. 物体运动的相对性和光速的绝对性, 两者之间的相互制约和作用乃是相对论里一切我们不熟悉的时空特征的根源.(部分参阅李新洲:《寻找自然之律--- 20世纪物理学革命》, 上海科技教育出版社, 2001.)1907 年德国数学家H. Minkowski (1864 ～1909) 提出了―Minkowski 空间‖,即把时间和空间融合在一起的四维空间1,3R. Minkowski 几何为Einstein 狭义相对论提供了合适的数学模型.“没有任何客观合理的方法能够把四维连续统分离成三维空间连续统和一维时间连续统. 因此从逻辑上讲, 在四维时空连续统(space- time continuum)中表述自然定律会更令人满意. 相对论在方法上的巨大进步正是建立在这个基础之上的, 这种进步归功于闵可夫斯基(Minkowski).”—Albert Einstein, The Meaning of Relativity, 1922, Princeton University Press. 中译本, 阿尔伯特·爱因斯坦著, 相对论的意义, (普林斯顿科学文库(Princeton Science Library) 1), 郝建纲、刘道军译, 上海科技教育出版社, 2001, p. 27.有了Minkowski 时空模型后, Einstein 又进一步研究引力场理论以建立广义相对论. 1912 年夏他已经概括出新的引力理论的基本物理原理, 但是为了实现广义相对论的目标, 还必须寻求理论的数学结构, Einstein 为此花了 3 年的时间, 最后, 在数学家M. Grossmann 的介绍下学习掌握了发展相对论引力学说所必需的数学工具—以Riemann几何和Ricci, Levi - Civita的绝对微分学, 也就是Einstein 后来所称的张量分析.“根据前面的讨论, 很显然, 如果要表达广义相对论, 就需要对不变量理论以及张量理论加以推广. 这就产生了一个问题, 即要求方程的形式必须对于任意的点变换都是协变的. 在相对论产生以前很久, 数学家们就已经建立了推广的张量演算理论. 黎曼(Riemann)首先把高斯(Gauss)的思路推广到了任意维连续统, 他很有预见性地看到了……进行这种推广的物理意义. 随后, 这个理论以张量微积分的形式得到了发展, 对此里奇(Ricci)和莱维·齐维塔(Tulio Levi-Civita, 1873～1941)做出了重要贡献. ”—阿尔伯特·爱因斯坦著, 相对论的意义, 郝建纲、刘道军译, 上海科技教育出版社, 2001, p. 57.从数学建模的角度看, 广义相对论讨论的中心问题是引力理论, 其基础是以下两个假设: 1. (等效原理)惯性力场与引力场的动力学效应是局部不可分辨的，(或说引力和非惯性系中的惯性力等效)；2. (广义相对性原理) 一切参考系都是平权的，换言之，客观的真实的物理规律应该在任意坐标变换下形式不变——广义协变性(即一切物理定律在所有参考系[无论是惯性的或非惯性的]中都具有相同的形式)。
Constructions of Mutually Unbiased Bases
Indeed, the right hand side equals 0 when κ = λ because the argument k + α ranges through all values of Fq ; and equals 1 when κ = λ. √ Note that all components of the sequence bλ,α have absolute value 1/ q , hence the basis Bα and the standard basis are mutually unbiased, for any α ∈ Fq . By computing the inner product | bκ,α , bλ,β | for α = β , we see that the terms cubic in k cancel out and, moreover, that the exponent is given by the trace of a quadratic polynomial in k . By Lemma 1 the inner product evaluates to q −1/2 , hence Bα and Bβ are mutually unbiased. 2 Remark 1. A remarkable feature of the previous construction is that knowledge of one basis Bα is sufficient because shifting the indices by adding a field element yields the other bases. The construction does not work in characteristic 2 and 3 because in these cases the sets Bα and Bβ , with α = β , are not mutually unbiased. Ivanovi´ c gave a fresh impetus to the field in 1981 with his seminal paper [14]. Among other things, he gave explicit constructions of p + 1 mutually unbiased bases of Cp , for p a prime. His construction was later generalized in the influential paper by Wootters and Fields [22], who gave the first proof of the following theorem. This proof was recently rephrased by Chaturvedi [9], and an alternate proof was given by Bandyopadhyay et al. [3]. We give a particularly short proof by taking advantage of Weil sums. Theorem 2. Let Fq be a finite field with odd characteristic p. Denote by Ba = {va,b | b ∈ Fq } the set of vectors given by