Generalized Space-time Supersymmetries, Division Algebras and Octonionic M-theory
Quantum Mechanics
Quantum MechanicsQuantum mechanics is a fascinating and complex field of physics that has revolutionized our understanding of the universe at the smallest scales. At its core, quantum mechanics deals with the behavior of particles at the quantum level, where the classical laws of physics break down and give way to a whole new set of rules. This field has given rise to many groundbreaking theories and technologies, such as quantum computing and quantum cryptography, that have the potential to revolutionize the way we live and interact with the world around us. One of the key principles of quantum mechanics is the concept of superposition, which states that a particle can exist in multiple states simultaneously until it is observedor measured. This idea challenges our classical intuition, which tells us that an object can only be in one place or state at a time. The famous thought experiment known as Schr?dinger's cat illustrates this concept, where a cat in a box is both alive and dead until the box is opened and the cat is observed. This idea of superposition has profound implications for the nature of reality and has led to many thought-provoking philosophical debates about the nature of existence. Another important concept in quantum mechanics is entanglement, where twoparticles become interconnected in such a way that the state of one particle is directly linked to the state of the other, regardless of the distance between them. This phenomenon, famously referred to as "spooky action at a distance" by Albert Einstein, challenges our understanding of causality and suggests that particlescan communicate instantaneously with each other, defying the limitations of space and time. This idea has been experimentally verified through a series of groundbreaking experiments and has opened up new possibilities for quantum communication and teleportation. The implications of quantum mechanics extend far beyond the realm of theoretical physics and have the potential to revolutionize technology in ways we can only begin to imagine. Quantum computing, for example, harnesses the principles of superposition and entanglement to perform calculations at speeds that far surpass classical computers. This has the potential to revolutionize fields such as cryptography, drug discovery, and artificial intelligence, unlocking new possibilities for innovation and discovery. Similarly, quantum cryptography uses the principles of quantum mechanics to create securecommunication channels that are theoretically impossible to hack, offering a new level of security and privacy in an increasingly digital world. Despite the incredible potential of quantum mechanics, there are still many challenges and mysteries that remain to be solved. The field is notoriously complex and counterintuitive, with many of its fundamental principles defying our classical understanding of the world. This has led to many debates and disagreements among physicists about the true nature of quantum mechanics and how best to interpretits implications. The famous Copenhagen interpretation, for example, posits that particles exist in a state of superposition until they are observed, while the many-worlds interpretation suggests that every possible outcome of a quantum event actually occurs in a separate parallel universe. These differing interpretations highlight the deep philosophical questions that quantum mechanics raises about the nature of reality and our place in the universe. In conclusion, quantum mechanics is a field that continues to push the boundaries of our understanding of the universe and challenge our most deeply held beliefs about the nature of reality. Its principles of superposition, entanglement, and uncertainty have revolutionized our understanding of the quantum world and opened up new possibilities for technology and innovation. While there are still many mysteries and debates surrounding quantum mechanics, its potential to revolutionize fields such as computing, communication, and cryptography is undeniable. As we continue to explore the implications of quantum mechanics, we are sure to uncover even more profound insights into the nature of the universe and our place within it.。
Godel-type space-time metrics
(6)
(7)
and hence an easy calculation gives 2 dM HM −1 = a (c σ3 − s σ1 ) + b [γ (c σ1 + s σ3 ) + σ ε] dr dM 2 HM −1 = a ε − b [γ − σ , (c σ3 − s σ1 )] dφ dM 2 HM −1 = a [γ ε + σ (c σ1 + s σ3 )] − b . dψ 3
G¨ odel -type space-time metrics
arXiv:gr-qc/0208093v3 14 Mar 2003
Antonio Enea Romano, Charles Goebel
Physics Department, University of Wisconsin, 1150 University Avenue, 53706 Madison, USA
This metric and its curvature are nonsingular at r = 0 if r and φ are regarded as polar coordinates, i.e., φ and φ + 2π specify the same point and r is nonnegative. It is the metric of the space GS which G¨ odel [1] (footnote 12) obtained by stretching by the factor µ a three-dimensional anti-deSitter space in the direction of a system of time-like Clifford parallels. [ For µ = 1 , hence b = 0 in Eq.(5), Eq.(9) is the metric of the adS space; upon multiplying the first term by µ2 and then redefining dt → dt/µ the metric Eq.(9) for general µ is obtained. ] The metric Eq.(9) was also obtained in [2] (in their Eq.(38) replace τ by t/2 and r by r/2 ) by what they call a ‘squash’, rather than a stretch. And finally, it is a special case of the metric (III.14) of [3], for e = −1 , F and G = constant . Using the metric Eq.(9) for the ds2 of Eq.(1), ds2 = [ dt + µ (γ − 1) dφ ]2 − σ 2 dφ2 − dr 2 − dz 2 (10)
外星文明猜想(英文作文)
Certainly! Here’s an essay exploring the conjectures about extraterrestrial civilizations, delving into the scientific, philosophical, and speculative aspects of the topic. Extraterrestrial Civilizations: The Great Beyond and Our Place in the CosmosThe universe, vast and ancient, stretches its arms across 93 billion light-years of observable space, containing billions of galaxies, each with billions of stars. Within this cosmic tapestry, the question of whether we are alone has captivated human minds for centuries. This essay explores the conjectures surrounding extraterrestrial civilizations, from the scientific theories to the speculative musings that fuel our imaginations.The Drake Equation: A Mathematical Framework for SpeculationAt the heart of the search for extraterrestrial intelligence (SETI) lies the Drake equation, formulated by astronomer Frank Drake in 1961. This mathematical framework attempts to estimate the number of active, communicative civilizations in the Milky Way galaxy. Variables include the rate of star formation, the fraction of stars with planetary systems, the number of planets capable of supporting life, the fraction of those planets where life actually emerges, the fraction of those life-bearing planets that develop intelligent life, the fraction of those that develop a civilization with technology, and the length of time such civilizations release detectable signals into space. While many of these variables remain unknown, the Drake equation serves as a tool for structured speculation and highlights the immense challenge in estimating the likelihood of extraterrestrial life.The Fermi Paradox: Where Are They?The Fermi paradox, named after physicist Enrico Fermi, poses a compelling question: Given the vastness of the universe and the high probability of habitable worlds, why have we not encountered any evidence of extraterrestrial civilizations? This paradox has led to numerous hypotheses. Perhaps civilizations tend to destroy themselves before achieving interstellar communication. Or, advanced civilizations might exist but choose to avoid contact with less developed species, adhering to a cosmic form of the “prime directive” seen in science fiction. Alternatively, the distances between stars could simply be too great for practical interstellar travel or communication, making detection exceedingly difficult.The Search for TechnosignaturesIn the quest for extraterrestrial intelligence, scientists have focused on detecting technosignatures—signs of technology that might indicate the presence of a civilization elsewhere in the universe. These include radio signals, laser pulses, or the dimming of stars due to megastructures like Dyson spheres. SETI projects, such as the Allen Telescope Array and Breakthrough Listen, scan the skies for anomalous signals that could be attributed to alien technology. While no definitive technosignatures have been found to date, the search continues, driven by advances in technology and a growing understanding of the cosmos.Astrobiology: Life Beyond EarthAstrobiology, the study of the origin, evolution, distribution, and future of life in the universe, offers insights into the conditions necessary for life. Research in astrobiology has revealed that life can thrive in extreme environments on Earth, suggesting that the conditions for life might be more widespread in the universe than previously thought. The discovery of exoplanets in the habitable zones of their stars, where liquid water can exist, increases the probability of finding environments suitable for life.Continued exploration of our solar system, particularly of Mars and the icy moons of Jupiter and Saturn, holds promise for uncovering signs of past or present microbial life. The Philosophical ImplicationsThe possibility of extraterrestrial civilizations raises profound philosophical questions about humanity’s place in the universe. Encountering another intelligence would force us to reevaluate our understanding of consciousness, culture, and ethics. It could lead to a new era of global unity as humanity comes together to face the challenges and opportunities of interstellar diplomacy. Conversely, it might also highlight our vulnerabilities and prompt introspection on our stewardship of the planet and our responsibilities as members of the cosmic community.Concluding ThoughtsWhile the existence of extraterrestrial civilizations remains a conjecture, the pursuit of answers has expanded our understanding of the universe and our place within it. The search for life beyond Earth is not just a scientific endeavor; it is a philosophical journey that challenges us to consider our origins, our destiny, and our role in the vast cosmic drama unfolding around us. Whether we find ourselves alone or part of a galactic community, the quest for knowledge about the universe and our place in it is one of humanity’s most enduring and inspiring pursuits.This essay explores various aspects of the conjectures surrounding extraterrestrial civilizations, from the scientific frameworks used to estimate their likelihood to the philosophical implications of their existence. If you have specific areas of interest within this broad topic, feel free to ask for further elaboration! If you have any further questions or need additional details on specific topics related to extraterrestrial life or astrobiology, please let me know!。
拉格朗日力学英文
拉格朗日力学英文Lagrangian MechanicsLagrangian mechanics is a formulation of classical mechanics that describes the motion of a system in terms of a function called the Lagrangian which is the difference between the kinetic and potential energies of the system. This approach provides a powerful and elegant way to derive the equations of motion for a wide range of physical systems, from simple pendulums to complex multi-body systems.The foundation of Lagrangian mechanics is the principle of stationary action which states that the motion of a system will follow the path that minimizes the action integral over the time interval of interest. This principle can be used to derive the Euler-Lagrange equations which are a set of second-order differential equations that describe the motion of the system. The Lagrangian function is defined as the difference between the kinetic energy and the potential energy of the system, and the Euler-Lagrange equations relate the Lagrangian function to the forces acting on the system.One of the key advantages of Lagrangian mechanics is its ability tohandle systems with constraints. Constraints are restrictions on the motion of the system, such as the motion of a pendulum being restricted to a circular path. In Lagrangian mechanics, these constraints are incorporated into the Lagrangian function, and the Euler-Lagrange equations automatically take them into account. This makes Lagrangian mechanics particularly useful for analyzing complex systems with multiple degrees of freedom and complicated constraints.Another advantage of Lagrangian mechanics is its ability to handle non-conservative forces, such as friction or damping. These forces can be incorporated into the Lagrangian function through the use of generalized coordinates and the principle of virtual work. This allows for the analysis of a wide range of physical systems, including those with dissipative forces.Lagrangian mechanics also provides a powerful tool for analyzing the symmetries of a system. Symmetries in the Lagrangian function can lead to conserved quantities, such as momentum or angular momentum, which can simplify the analysis of the system's motion. This is particularly useful in fields such as particle physics, where the underlying symmetries of the system play a crucial role in determining the behavior of the particles.The formulation of Lagrangian mechanics was developed by theFrench mathematician Joseph-Louis Lagrange in the late 18th century. Lagrange's work built upon the earlier work of Euler and Hamilton, and it provided a more general and unified framework for describing the motion of mechanical systems. Lagrangian mechanics has since become a fundamental tool in the study of classical mechanics and has been extended to other areas of physics, such as field theory and quantum mechanics.In Lagrangian mechanics, the motion of a system is described by a set of generalized coordinates which can be any set of independent variables that uniquely specify the configuration of the system. The Lagrangian function is then defined in terms of these generalized coordinates and their time derivatives. The Euler-Lagrange equations are then used to derive the equations of motion for the system, which can be solved to determine the trajectory of the system over time.One of the key applications of Lagrangian mechanics is in the analysis of multi-body systems, such as the motion of planets in the solar system or the motion of robots with multiple joints. In these systems, the Lagrangian function can be used to derive the equations of motion for the entire system, taking into account the interactions between the various components. This makes Lagrangian mechanics a powerful tool for the design and analysis of complex mechanical systems.Another important application of Lagrangian mechanics is in the field of control theory, where it is used to design control systems that can manipulate the motion of a system in a desired way. By using the Lagrangian function to describe the system's dynamics, control engineers can develop control algorithms that can efficiently and effectively control the motion of the system.Overall, Lagrangian mechanics is a powerful and versatile approach to the study of classical mechanics that has had a profound impact on the field of physics and engineering. Its ability to handle complex systems with constraints and non-conservative forces, as well as its connection to symmetries and conserved quantities, make it a essential tool in the study of a wide range of physical phenomena.。
The origin of space-time as seen from matrix model simulations：时空的起源,从矩阵模型模拟
Gauge/gravity duality predicts that this should be reproduced by 1d SYM. large-N, low T
microscopic origin of the black hole thermodynamics
quantum description of the states inside the BH
Comparison including corrections
Hanada-Hyakutake-J.N.-Takeuchi, PRL 102 (’09) 191602 [arXiv:0811.3102]
corrections
3. Direct test of gauge-gravity correspondence
Matrix model formulation of superstring/M theories
Banks-Fischler-Shenker-Susskind (2019), Ishibashi-Kawai-Kitazawa-Tsuchiya (2019)
Dynamical origin of space-time Applications to the physics beyond the Standard Model
interactions between D-branes string field theory from SD eqs. for Wilson loops
Fukuma-Kawai-Kitazawa-Tsuchiya (’98) c.f.) Matrix Theory Banks-Fischler-Shenker-Susskind (’96)
电动机和磁悬浮技术相关内容说明书
IndexAC motor,461active magnetic bearings,1,10 actuator,111,152 electrostatic,488gain,117measuring,131micro magnetic,487 model,330model assembly,117 response limitations,127stiffness,117voice coil,495actuator offset,mechanical,187 aerodynamic losses,136,140 aeroengine,279aerospace,7air drag losses,159algorithmlevitation control,467P+2,467P-2,467aliasing,236,245alloyscobalt,93AMB system model,328Amp´e re’s loop law,115Amp´e re’s law,72amplifier,112analog,97losses,148operating modes,126 power,69,77,97 switching,97,450transconductance,121 transpermance,122analogcontrol,229,231,233 electronics,229filter,245hardware,236analog-to-digital A/Dconversion channel,230 conversion resolution,231,233,245 conversion time,231,238 converter,229,230,233,234,246 anti-aliasingfilter,231,236,333 applications of AMB,17arithmeticsfixed-point,220floating point,245integer,220,245,246artificial heartimplantable,480pump,462,480artificial heart pump,17automatic balancing,426auxiliary bearing,389,407,412,513 contact,407,410,412–421,423,424 contact modes,413–421,423,424 friction,413,415,419,424 touchdown recovery,410,427,431 axialself-bearing motor,477axis of geometry,215back-up bearing,389524Indexbackward difference,239,242 backward whirl,390,396,413,414,424 balancingactive,516automatic,426bandwidth,320,321power bandwidth,153base motion,409Beams,Jesse,499bearingauxiliary,407,412ball,475combined,motor,461elastic suspension,260forces,171,173homopolar,140,148load capacity,81PM repulsion,477stiffness,153,173thrust,magnetic,93bearingless motor,461bi-quad representation,246biascurrent,31–33,35,41,79,224flux,28linearization,79,95,440,443 permanent magnet,95,468 bismuth,496blade loss,409braking torque,135,144cylinder,141disc,141measurement,146cable losses,138,148Campbell diagram,207,212 capacitive displacement sensor,103 casing model,339center of gravity control,361central difference,243chaotic motion,390characteristic polynomial,35,61 characteristics of AMB,15circuitmagnetic,74classification of AMB,10closed loop model,341cobalt alloys,93coefficientdrag,141,144influence,420coercivefield intensity,74coilconfiguration,411design,82,88temperature,88winding scheme,90collocated,199,203,204non-,194,199,200,203,208 collocation,437combined motor bearing,461 compliancedynamic,66compressorslosses in,149conductor,71conicalmode,198,199,206,208,210–212, 214motion,198continuous-time,233control,237,240,243–245differential equation,233 eigenvalue,235,240equivalent,243frequency variable,236plant,233,234,238signal,239system,234–238,240,243 control,29,33,152H∞,37,52,57,61,214,242,367μ,370axial,220bandwidth,41,205,211,321 center of gravity,361COG coordinate,210–212 complexity,383conical mode,211,212,214,224 current,49–52,193–195,224 decentralized,342decentralized/local,194–197,199, 203–208,210,212 decoupled,208,211,212,224 design,33,34,37,52,54,193,208, 215digital,29,38,50,57,65,220 digital PID,471fault tolerant,514Index525flexible rotor,194,215force,34,37,207,209gain,65gain compensation,322gain scheduled,377harmonic,426levitation,467linear,31,34,53,54LPV,377LQ,243LQ/LQG,54,56,214LQG,441MIMO(multi-channel),30,52,65, 204,208,219minimal energy,160Mixed PID,361modal,209,210moment of force,209non-collocated PID,352order,57parallel mode,211,212,214,224 passive,28,57,375PD,418,425,467,479PD/PID,39,42,44–46,57,194,196, 199,205–208,213,237,239–245 phase,65phase compensation,322phase lag,47phase lead,321PID,47,342,411,413,417pole-placement,54,56,214,243rigid body,194,199,214robust,42,57,61,513roll off,205SISO(single-channel),30,51,65,208, 219state estimator/observer,214state space,52,53,60 synchronous,378,426,427 synchronous current,220,223 synchronous displacement,220 synchronous force,221,222,224 system,69μ−synthesis,37,52,57,61,214,242 tilt and translate,361unbalance,215–217,219,220,224, 378,426underlying current,49,50 underlying force,51voltage,49,50,53,54,224 controllerdesign,319cooling,81,151,158coordinatesbearing,193center of gravity/mass(COG),193, 194,196,211,212sensor,193,209copper losses,137copper resistance,87corrective procedures,516couplingA-B,208,211cross-,208,211,214de-,211coupling effects,174cracked rotor,410critical speed,167,182,183 bending,215,217,218rigid body,216,220,221currentmeasurement,105phase,475sheet,466,468,470damping,28,29,34,36,39,61,65,66, 153,199,206,212,337critical,36,43cross-,203external,204,259inner,203matrix,196,201,203,210,213“natural”,196,208nutation,207overcritical,36synchronous,216undercritical,36dead time,238decentralized control,342 decomposition,199,210degree of freedom(DOF),28,30,51, 52,59,65,191,196,208,213 delay,333computation,231,245,246 sampling,238–241,243–246time,230,231,245densitygas,141526Indexdependability,507design,147coil,88limitations,151magnets,81quality,510software,510systematic checks,510thrust magnetic bearings,93touch-down bearing,401 destabilization,199,201,203,208,214 diagnosis,316,515active,515diamagnetic materials,6,495difference equation,233differentialdriving mode,linearization,80 sensing,101differential equation,59,195,210 closed-loop,35,212first-order,52,53homogeneous,36,55 inhomogeneous,58matrix,193,194,196,201,210,213 second-order,53state space,213vector,52differential winding,124digitalcontrol,229,231,233,237,245 control design,243control,PID,471filter,231,245hardware,230,231signal processor,467,472,478,482 digital signal processor(DSP),229,230, 232,245,247,248digital-to-analog D/Aconversion resolution,231 conversion time,231converter,229–231,233,234,236,245 discrete-time,233control,233,236–241,244,245 eigenvalue,235,240equivalent,243filter,241frequency response,236frequency variable,236plant,237,244system,231,233,235,236,240 transfer function,236,241diskrigid model,413displacementvirtual,78dither,generalized,443dragviscous,140drag coefficient,141,144shrouded cylindrical rotor,142droprotor,412dynamiccompliance,66stiffness,28,34,46,63,66 dynamicsrigid rotor,167dynamic stiffness,153Earnshaw’s theorem,5eccentricity,169eddy current losses,135,137,139,159 eddy currents,14,84,130,452 sensor,101,482eigendamping,37,41 eigenfrequencies,30,37,41,59,171 closed-loop,205,208gyroscopic effects,176nutation,207,212rigid body,205,207,208,212 eigenmode,55backward,198bending,205closed-loop,205conical,198,204,206,207 coupling,200decomposition,198forward,204,206nutation,207,212parallel,198,204precession,198rigid body,205,206,209 eigenvalues,36,50,53,55,58,61,196, 198,206,235–237,240,246,320 closed-loop,34,36,40,41,196,198, 199conjugate complex,35open-loop,33,35,49,205,208Index527real,36,201trajectory,196,197,199,212 electro-dynamic levitation,14 electromagnet,27–30,32,35,44,69, 115inductance,436elevator guideways,435,442energymagnetic,489equations of motionflexible rotor,272estimationparameter,447Euler angles,191Euler-Bernoulli beam model,337 exampleH∞control,369actuator model,120asymmetric rotor,375center of gravity control,362 mixed PID control,362non-collocated PID control,352 PID control,344PID performance analysis,351 rotor sensitivity,360sensitivity analysis,357system model,336tilt and translate control,362 excitation,182backward whirl,187external,58force,28,58,59forward whirl,185frequency,58,59harmonic,55mechanical sources,187node,62non-periodic,188parametric,188periodic,55sensor and actuator offset,187 unsymmetries of the rotor,188 factorforce-current,k i,79force-displacement,k s,79fail-safe,513failure modes,411failures of AMB,508,513Faraday’s law,115,436fault detector,411fault tolerance,407faults,516AMB system,408rotor,409,410feedback,34gain,41,42integrating,45,47output,54,56,57state,54,56,214velocity,43,47ferromagnetic,28,35ferromagnetic materials,6,73,495fieldmagnetic,71filteranti-aliasing,333Finite Element Method,251,267 model reduction,292finite element modeling,82flexibilityrotor,319flexiblemode shapes,rotor,324flexible rotor,155,191,193,194,203, 205,208,215,251,263 equations of motion,272with AMB,288fluid bearingidentification,312,313fluid structure interaction,312fluxdistribution,465leakage,88measurement,105flywheels,17losses in,149forcelevitation,465Lorentz,70,473,494magnetic,77,152 magnetomotive,75maximum,152specific,489force-free,223,224force/currentfactor,33,45,48,192matrix,210528Indexrelationship,31,45force/displacementfactor,33,192matrix,193relationship,31,33forced vibrations,256aeroengine,285response,262unbalance,284forcesbearing,171nonconservative,167,174,179 forward whirl,414,419Fourier/frequency analyzer,247 FPGA,231free rotor,175free-free mode shapes,324frequency domain,57,59,61 frequency response,34,52,55,60–63, 223,236,237,241,243,244 amplification,58–60amplitude,59–61identification,302matrix,61measurement,247,249phase,60,61unbalance,185friction,413,415,419,424gain compensation,322gain margin,356gain scheduled control,377gap sensor,467gas density,141gas friction losses,136,140graphitepyrolithic,496gyrodynamics,176gyroscopic effects,173,176elastic rotor,274gyroscopics,28,65,231,242,247,248, 377effect,191,198,200,204,208,211 matrix,194,196,198,199,201,213 rotor,202H-bridge,98Hall effect,105current measurement,107hallbach array,497harmonic balance,420harmonic control,426heartartificial,pump,462,480 implantable,artificial,480 transplant,480heat loss,84heteropolar,159heteropolar magnetics,82high speed,7high speed rotor,154high temperature,158 homopolar,159homopolar bearing,140,148 homopolar magnetics,83,97hybridmagnetic bearing,468 hysteresis,74,151,491losses,159hysteresis losses,137,138identification,229,247,252,299,516 for diagnosis,316excitation by AMB,305fluid structure interaction,312 parameter estimation,304 response functions,302 implantable artificial heart,480 impulse responseidentification,302inductance,72,76 electromagnet,436inertia properties,167influence coefficient,420 information processing,152initial condition,36,52 instrumentation,39built-in,230,246,247external,247,248integratorgain,PID controller,346inter-sample skew,230,231 interlacingdefect repairing,326pole-zero,323interrogation signal,446,450,454 iron resistivity,139ISO standardsIndex529for AMB,509quality,509sensitivity,358unbalance,181Jeffcott rotor,252kinematic viscosity,141lamination,139Laval rotor,252leakageflux,88levitationcoil current,470control,467control algorithm,467force,465Levitron,5lifetime at high temperature,158 LIGA,487linearperiodic,443time invariant,439linearity/nonlinearity,31,33,41,43,48, 50,51,57linearization,28,33,44,46,48,49,124 bias,95current bias,79square root,126load capacity,39,44,47,67,81,151 radial bearings,92specific,92thrust bearing,94loadscentrifugal,491torque,467Lorentzforce,473self-bearing motor,462,473Lorentz force,10,13,70,494loss mechanisms,135losses,135,159aerodynamic,140,147amplifier,148cable,138,148copper,84,137eddy current,135,137,139 electrical power,95gas friction,140hysteresis,137,138iron,84,147magnetic,136mitigation,147power amplifier,138rotational,492stator,148windage,135,140,501low passfilter,205,215,239,241,242, 245LPV control,377LQG control,440,441Lyapunovfunction,448machinesmart,411MAGLEV,6,14magnetpermanent,496rare earth,496magneticcircuit,74field,71field energy,77flux,71flux density,71force,77permeability,72polarization,73saturation,81,92,435,453 magnetic actuator,111magnetic bearingactive,27–29,37,47active micro,498hybrid,29,468Lorentz force,27passive,27–29,37reluctance force,27 superconducting,27types,493magnetic displacement sensor,103 magneticfluxload capacity,151magnetic force,10magnetic loss,136magnetism,70magnetization curve,76 magnetomotive force,75530Indexmaintainability,410margin,gain and phase,356massmatrix,194,201,203,213rotor,205materialscarbonfiber,155cobalt,152diamagnetic,6,495 ferromagnetic,6,10,151,495for high temperature,158 strength,94superconducting,12,495 maximum singular value,350 measurementforce,306mechanical energyconversion,48kinetic,52potential,52mechatronics,39,135definition,4MEMS,487micro magnetic actuator,487 microprocessor,229–231,245,247,248fixed-point,220MIMOcontrol,230,231,246control design,241–243,245,247 measurement,247transfer function,247,248modal analysis,252for rotating structures,307modal parameters,300modal truncation,337mode shapesflexible rotor,324free-free,324modelactuator,330AMB system,328assembly,335casings and substructures,339 closed loop,341Euler-Bernoulli beam,337rotor,331sensor,332state space,113,329structure,330synchronously reduced,342 modelingfinite element,82modes,337modulationpulse-width,97moment of inertiapolar,206transverse,205,206monitoring,229,230,247,248motorAC,461,468axial self-bearing,477 bearingless,461combined,bearing,461 induction,466,500Lorentz self-bearing,462,473self-bearing,461,479self-bearing or bearingless,27 motor control,230multi-processor,230multiplexer,230,246multiply-accumulate(MAC),246 natural vibrations,256,262 aeroengine,279flexible rotor,277networkthermal,85non-collocation,320,323non-conservative forces,167,179,203, 214matrix,213,214nonlinearmagnetic analysis,76nonlinear dynamicsamplitude jump,428aperiodic response,415contact modes,413–421,423,424 rotordynamics,413–421,423,424 normvector,349H∞,366notchfiltergeneralized,216,219,220 nutation,167,178Nyquist frequency,236–238Ohm’s law,436Index531operating point,31,34,35,45,48 optical displacement sensor,104 optimizationbearing geometry,88order reduction,246,373oscillationamplitude,37,58damped,36,37,43harmonic,37,58periodic,37phase,58pseudo-periodic,37 oscilloscope,247,248over-sampling,231Pad´e approximation,333Pad´e approximation,245parallelmode,198,199,204,206,208,210–212,214paramagnetic material,73 parameter estimation,304 parameter estimator,447parametric excitation,188PD control,321,418,425,467,479 performanceassessment,347peripheral,229–231permanent magnets,5,12,28,29,31, 95bias,468permeability,10persistency of excitation,447,450 phasecurrents,475phase compensation,322phase laganti-aliasingfilter,236sampling delay,237–239,244 phase lead controller,321phase margin,238,356PIDcontrol,digital,471PID control,342,411,413,417non-collocated,352pinned beam,320plantMIMO,208rigid body,208polarity sequence,140polarizationmagnetic,73poles,320angle,78number,463pair number,464plus two or minus two,462 salient,466–468shoes,88transfer function,323positionsensor,99position reference command,44–46 position sensor,69,435powerspecific,489power amplifier,27,29,41,42,44,50, 69,77,97,153bandwidth,50,56current,29,44,48,49dynamics,30,34,44,57voltage,29,48power amplifier losses,138power bandwidth,153power failure,410power supply,513precession,167,178precision bearings,161preventive maintenance,229,248 principal/inertial axis,215,223,224 probeproximity,478processordigital signal,467,472,478,482 properties of AMB,15proximityprobe,478pulse width modulation(PWM),97, 224signal pattern,230unit,229,231pumpartificial heart,462,480 performance,482rotary blood,480real time operating system,510 recomposition,209532Indexredundancy,408–411,511 reliability,505reluctance force,10remanence,74resistancecopper,87resistivityiron,139resonance,182rigid body,221responseunbalance,351,467,468,472,475, 479response functionsidentification,302 measurement,303retainer bearing,389Reynolds number,140rigid disk model,413rigid rotor,320equations of motion,171rigid rotor/body,191,193,194,196, 200,204,208,210,212,214 robust control,513robustness,34,50,56,57,61,63,356 root locus,325rotaryblood pump,480rotationforce-free,160high speed,7high-speed,154precision,161rotational losses,492rotor,29-stator contact,33,62drop,412excitation,182flexibility,319flexible,155,251,263free,175model,331modes,337non-rotating,175position,29,30,39,46,47rigid,320rigid,dynamics,167velocity,52rotor lossesmeasurement,146rotor responsebackward whirl,413,419contact modes,413,418,424 forward whirl,414rotor speedgyroscopic effects,176influence on eigenfrequencies,178 rotor/stator contact,390rub,410,414,419,420,423,424,431 safety,505definitions,507philos.background,506salient pole,466–468sample-and-hold,230,234sampling period,231,233,234,239,241 sampling rate/frequency,231,233,236, 237,244–246saturation,74,151dynamic,205magnetic,81,435,453numerical,220power amplifier,223scaling,489,490secant algorithm,239,242,244self monitoring,411self–sensing,435self-bearing motor,27,461,479 sensitivity,355ISO standards,358sources of,361sensitivity function,62,63sensor,27,41,42capacitive,displacement,103 displacement,position,411 dynamics,30,34,44,57eddy current,29,101,482gap,467inductive,29inductive,displacement,101 magnetic displacement,103model,332noise,56optical,3optical displacement,104position,69,99transverseflux,102sensor offset,mechanical,187Index533Shannon theorem,236,245signalinterrogation,446,450,454signal generator,247signal injection,247singular value,350SISOcontrol,230,241transfer function,247size of the bearing,158skew-symmetric,194,201,203,211 small gain theorem,356smartdefinition,514examples,514machines,514smart machine,411smart machine technology,161 software,510as a machine element,4 development system,510specific force,489specific load capacity,152specific power,489speedsupercritical,155spring-damper,28,34,37,49 mechanical,30,31,34,44square root linearization,126 stability margins,gain and phase,356 stability of motion,171,174stability/instability,28,31,33,35 asymptotic,58,61closed-loop,41,57limit-stable,36open-loop,30,35,49,50 stabilization,29,33standards,509for magnetic bearings,9ISO,sensitivity,358state spacedescription,196,214matrix,199state space model,113,329statorslotless,474slotted,474stiffness,30,34,36,39,42,44,61,65, 66,199closed-loop,39,42compensation matrix,211,212,224 coupling,214cross-,201,203dynamic,28,34,46,63,66,153 external,222high,41,42low,40,50matrix,196,201,203,210,213 mechanical,31“natural”,42,196,208negative bearing,31,33,40,42,45, 193,194,211,223,224static,28,34,42,47,63,67strengthmaterial,94stresses,centrifugal forces,154 substructure model,339 superconducting materials,495 superconductivity,12,14,18 superconductor,495suspensionactive,29five axes,29passive,28permanent magnet,29rigid body,30switchingamplifier,97,446,450noise,97ripple,446,450three state,451symmetric,194,198,201,203–205,208, 211,212non-,199,212synchronous control,378,426,427 synchronously reduced model,342 synthesisμ,370system model,328technology of AMB,9temperaturecoil,88test rigaeroengine,279for identification,312534Indexfor modal analysis,308for touch-down,391thrust magnetic bearings,93tilt and translate control,361tilting motion,198time domain,36,55,59–61timer interrupt,232torquebraking,135,144dynamic,479load,467reaction,473rotating,468,471touch-down bearing,389ball bearings,400contact dynamics,391contact force,399control reconfig.,518design guidelines,401friction force,395test rig,391time history of touch-down,399,400 vertical axis arrangement,403whirl motion,396touchdown,407,413avoiding,407,426dynamics,412recovery,410,427,431 transconductance amplifier,112,121 transfer function,60–64,211,219,235, 236,241,247,248matrix,60,219poles and zeros,323transformFFT,61Fourier,55,61Laplace,59,60transformationZ,236bilinear,243Fast Fourier(FFT),247Laplace,235matrix,193,209–211natural coordinates,365Tustin,243transitionmatrix,235sampling instant,234,235state,233translational motion,211 transpermeance amplifier,122 transplantheart,480transportation,6transverseflux sensor,102turbo–molecular pump,435,444turbo-machinery,17turbomolecular pumps,149 unbalance,38,256,409,413,414,420 adaptation,216,218 compensation,516definition,180excitation,55excitation of rotor vibr.,183force attenuation,28,216forced vibrations,284forward whirl,185quality grade,181residual,215,216response,215,220,222,224,262,468, 472,475,479technical example,169vibration attenuation,216 unbalance control,229,378,426 unbalance response,351,467unit circle,235,236unsymmetry of the rotor,188 vacuum,501vacuum applicationslosses in,149velocity,34,43,48,52,53,239velocity measurement,104vertical axis,403vibrational control,443vibrationsbending,461forced,256,284,285natural,167,171,175,256,262,277, 279steady state,475virgin magnetization curve,84virtual displacement,78viscositykinematic,141viscous drag,140voice coil actuator,495Index535voltage control,439whirlbackward,413,414forward,414whirl motion,396whirl,forward and backward,167,178 windage losses,135,136,140,501windingdifferential,124zero power control,160zero-order hold(ZOH),233,234,238, 239,243zerostransfer function,323。
爱因斯坦相对论时间空间质量维度
In 1905, a mind-blowing idea burst onto the scene like fireworks on the Fourth of July: Albert Einstein's theory of relativity! This mind-bending theory shook up the way we think about timeand space, showing that they're not just separate things, but actually part of a wild four-dimensional ride called spacetime. According to Einstein, time and space aren't set in stone but can twist and turn depending on how you're looking at them. It'slike a cosmic rollercoaster, where time can stretch out and space can squish down depending on how fast you're zooming around. And get this - Einstein also blew our minds by saying that the speed of light is the same for everyone, no matter how fast they're going! This theory didn't just change the game - it flipped the whole universe on its head! So buckle up, becausewe're in for a wild ride through the wacky world of relativity!1905年，像国庆节的烟火一样，一个令人心碎的想法冲向现场：阿尔伯特·爱因斯坦的相对论！这个思维的理论震撼了我们对时间和空间的思考方式，表明它们不仅仅是独立的事物，而是一个叫做"时空"的四维飞行的一部分。
光线弯曲 时间膨胀
光线弯曲时间膨胀英文回答：Light Bending and Time Dilation.Light bending and time dilation are two importanteffects predicted by Einstein's theory of generalrelativity. Light bending refers to the deflection of light as it passes through a gravitational field, while time dilation refers to the slowing down of time in agravitational field.Both of these effects have been experimentally verified. Light bending was first observed during the solar eclipseof 1919. Time dilation has been observed in a variety of experiments, including experiments with atomic clocks.Light Bending.Light bending is caused by the curvature of spacetime.Spacetime is a four-dimensional fabric that is warped by the presence of mass and energy. Light travels along the shortest path through spacetime, so when light passes through a gravitational field, it follows a curved path.The amount of light bending depends on the strength of the gravitational field. The stronger the gravitational field, the greater the amount of light bending.Time Dilation.Time dilation is caused by the fact that time runs slower in a gravitational field. This is because the gravitational field makes it more difficult for objects to move. As an object moves slower, its clock runs slower.The amount of time dilation depends on the strength of the gravitational field. The stronger the gravitational field, the greater the amount of time dilation.Applications of Light Bending and Time Dilation.Light bending and time dilation have a number of applications in astronomy and cosmology. For example, light bending is used to study the distribution of mass in the universe. Time dilation is used to study the evolution of the universe.中文回答：光线弯曲和时间膨胀。
Laser Ranging to the Moon, Mars and Beyond
a r X i v :g r -q c /0411082v 1 16 N o v 2004Laser Ranging to the Moon,Mars and BeyondSlava G.Turyshev,James G.Williams,Michael Shao,John D.AndersonJet Propulsion Laboratory,California Institute of Technology,4800Oak Grove Drive,Pasadena,CA 91109,USAKenneth L.Nordtvedt,Jr.Northwest Analysis,118Sourdough Ridge Road,Bozeman,MT 59715USA Thomas W.Murphy,Jr.Physics Department,University of California,San Diego 9500Gilman Dr.,La Jolla,CA 92093USA Abstract Current and future optical technologies will aid exploration of the Moon and Mars while advancing fundamental physics research in the solar system.Technologies and possible improvements in the laser-enabled tests of various physical phenomena are considered along with a space architecture that could be the cornerstone for robotic and human exploration of the solar system.In particular,accurate ranging to the Moon and Mars would not only lead to construction of a new space communication infrastructure enabling an improved navigational accuracy,but will also provide a significant improvement in several tests of gravitational theory:the equivalence principle,geodetic precession,PPN parameters βand γ,and possible variation of the gravitational constant G .Other tests would become possible with an optical architecture that would allow proceeding from meter to centimeter to millimeter range accuracies on interplanetary distances.This paper discusses the current state and the future improvements in the tests of relativistic gravity with Lunar Laser Ranging (LLR).We also consider precision gravitational tests with the future laser rangingto Mars and discuss optical design of the proposed Laser Astrometric Test of Relativity (LATOR)mission.We emphasize that already existing capabilities can offer significant improvements not only in the tests of fundamental physics,but may also establish the infrastructure for space exploration in the near future.Looking to future exploration,what characteristics are desired for the next generation of ranging devices,what is the optimal architecture that would benefit both space exploration and fundamental physics,and what fundamental questions can be investigated?We try to answer these questions.1IntroductionThe recent progress in fundamental physics research was enabled by significant advancements in many technological areas with one of the examples being the continuing development of the NASA Deep Space Network –critical infrastructure for precision navigation and communication in space.A demonstration of such a progress is the recent Cassini solar conjunction experiment[8,6]that was possible only because of the use of Ka-band(∼33.4GHz)spacecraft radio-tracking capabilities.The experiment was part of the ancillary science program–a by-product of this new radio-tracking technology.Becasue of a much higher data rate transmission and, thus,larger data volume delivered from large distances the higher communication frequency was a very important mission capability.The higher frequencies are also less affected by the dispersion in the solar plasma,thus allowing a more extensive coverage,when depp space navigation is concerned.There is still a possibility of moving to even higher radio-frequencies, say to∼60GHz,however,this would put us closer to the limit that the Earth’s atmosphere imposes on signal transmission.Beyond these frequencies radio communication with distant spacecraft will be inefficient.The next step is switching to optical communication.Lasers—with their spatial coherence,narrow spectral emission,high power,and well-defined spatial modes—are highly useful for many space applications.While in free-space,optical laser communication(lasercomm)would have an advantage as opposed to the conventional radio-communication sercomm would provide not only significantly higher data rates(on the order of a few Gbps),it would also allow a more precise navigation and attitude control.The latter is of great importance for manned missions in accord the“Moon,Mars and Beyond”Space Exploration Initiative.In fact,precision navigation,attitude control,landing,resource location, 3-dimensional imaging,surface scanning,formationflying and many other areas are thought only in terms of laser-enabled technologies.Here we investigate how a near-future free-space optical communication architecture might benefit progress in gravitational and fundamental physics experiments performed in the solar system.This paper focuses on current and future optical technologies and methods that will advance fundamental physics research in the context of solar system exploration.There are many activities that focused on the design on an optical transceiver system which will work at the distance comparable to that between the Earth and Mars,and test it on the Moon.This paper summarizes required capabilities for such a system.In particular,we discuss how accurate laser ranging to the neighboring celestial bodies,the Moon and Mars,would not only lead to construction of a new space communication infrastructure with much improved navigational accuracy,it will also provide a significant improvement in several tests of gravitational theory. Looking to future exploration,we address the characteristics that are desired for the next generation of ranging devices;we will focus on optimal architecture that would benefit both space exploration and fundamental physics,and discuss the questions of critical importance that can be investigated.This paper is organized as follows:Section2discusses the current state and future per-formance expected with the LLR technology.Section3addresses the possibility of improving tests of gravitational theories with laser ranging to Mars.Section4addresses the next logical step—interplanetary laser ranging.We discuss the mission proposal for the Laser Astrometric Test of Relativity(LATOR).We present a design for its optical receiver system.Section5 addresses a proposal for new multi-purpose space architecture based on optical communica-tion.We present a preliminary design and discuss implications of this new proposal for tests of fundamental physics.We close with a summary and recommendations.2LLR Contribution to Fundamental PhysicsDuring more than35years of its existence lunar laser ranging has become a critical technique available for precision tests of gravitational theory.The20th century progress in three seem-ingly unrelated areas of human exploration–quantum optics,astronomy,and human spaceexploration,led to the construction of this unique interplanetary instrument to conduct very precise tests of fundamental physics.In this section we will discuss the current state in LLR tests of relativistic gravity and explore what could be possible in the near future.2.1Motivation for Precision Tests of GravityThe nature of gravity is fundamental to our understanding of the structure and evolution of the universe.This importance motivates various precision tests of gravity both in laboratories and in space.Most of the experimental underpinning for theoretical gravitation has come from experiments conducted in the solar system.Einstein’s general theory of relativity(GR)began its empirical success in1915by explaining the anomalous perihelion precession of Mercury’s orbit,using no adjustable theoretical parameters.Eddington’s observations of the gravitational deflection of light during a solar eclipse in1919confirmed the doubling of the deflection angles predicted by GR as compared to Newtonian and Equivalence Principle(EP)arguments.Follow-ing these beginnings,the general theory of relativity has been verified at ever-higher accuracy. Thus,microwave ranging to the Viking landers on Mars yielded an accuracy of∼0.2%from the gravitational time-delay tests of GR[48,44,49,50].Recent spacecraft and planetary mi-crowave radar observations reached an accuracy of∼0.15%[4,5].The astrometric observations of the deflection of quasar positions with respect to the Sun performed with Very-Long Base-line Interferometry(VLBI)improved the accuracy of the tests of gravity to∼0.045%[45,51]. Lunar Laser Ranging(LLR),the continuing legacy of the Apollo program,has provided ver-ification of GR improving an accuracy to∼0.011%via precision measurements of the lunar orbit[62,63,30,31,32,35,24,36,4,68].The recent time-delay experiments with the Cassini spacecraft at a solar conjunction have tested gravity to a remarkable accuracy of0.0023%[8] in measuring deflection of microwaves by solar gravity.Thus,almost ninety years after general relativity was born,Einstein’s theory has survived every test.This rare longevity and the absence of any adjustable parameters,does not mean that this theory is absolutely correct,but it serves to motivate more sensitive tests searching for its expected violation.The solar conjunction experiments with the Cassini spacecraft have dramatically improved the accuracy in the solar system tests of GR[8].The reported accuracy of2.3×10−5in measuring the Eddington parameterγ,opens a new realm for gravitational tests,especially those motivated by the on-going progress in scalar-tensor theories of gravity.1 In particular,scalar-tensor extensions of gravity that are consistent with present cosmological models[15,16,17,18,19,20,39]predict deviations of this parameter from its GR value of unity at levels of10−5to10−7.Furthermore,the continuing inability to unify gravity with the other forces indicates that GR should be violated at some level.The Cassini result together with these theoretical predictions motivate new searches for possible GR violations;they also provide a robust theoretical paradigm and constructive guidance for experiments that would push beyond the present experimental accuracy for parameterized post-Newtonian(PPN)parameters(for details on the PPN formalism see[60]).Thus,in addition to experiments that probe the GR prediction for the curvature of the gravityfield(given by parameterγ),any experiment pushingthe accuracy in measuring the degree of non-linearity of gravity superposition(given by anotherEddington parameterβ)will also be of great interest.This is a powerful motive for tests ofgravitational physics phenomena at improved accuracies.Analyses of laser ranges to the Moon have provided increasingly stringent limits on anyviolation of the Equivalence Principle(EP);they also enabled very accurate measurements fora number of relativistic gravity parameters.2.2LLR History and Scientific BackgroundLLR has a distinguished history[24,9]dating back to the placement of a retroreflector array onthe lunar surface by the Apollo11astronauts.Additional reflectors were left by the Apollo14and Apollo15astronauts,and two French-built reflector arrays were placed on the Moon by theSoviet Luna17and Luna21missions.Figure1shows the weighted RMS residual for each year.Early accuracies using the McDonald Observatory’s2.7m telescope hovered around25cm. Equipment improvements decreased the ranging uncertainty to∼15cm later in the1970s.In1985the2.7m ranging system was replaced with the McDonald Laser Ranging System(MLRS).In the1980s ranges were also received from Haleakala Observatory on the island of Maui in theHawaiian chain and the Observatoire de la Cote d’Azur(OCA)in France.Haleakala ceasedoperations in1990.A sequence of technical improvements decreased the range uncertainty tothe current∼2cm.The2.7m telescope had a greater light gathering capability than thenewer smaller aperture systems,but the newer systemsfired more frequently and had a muchimproved range accuracy.The new systems do not distinguish returning photons against thebright background near full Moon,which the2.7m telescope could do,though there are somemodern eclipse observations.The lasers currently used in the ranging operate at10Hz,with a pulse width of about200 psec;each pulse contains∼1018photons.Under favorable observing conditions a single reflectedphoton is detected once every few seconds.For data processing,the ranges represented by thereturned photons are statistically combined into normal points,each normal point comprisingup to∼100photons.There are15553normal points are collected until March2004.Themeasured round-trip travel times∆t are two way,but in this paper equivalent ranges in lengthunits are c∆t/2.The conversion between time and length(for distance,residuals,and dataaccuracy)uses1nsec=15cm.The ranges of the early1970s had accuracies of approximately25cm.By1976the accuracies of the ranges had improved to about15cm.Accuracies improvedfurther in the mid-1980s;by1987they were4cm,and the present accuracies are∼2cm.One immediate result of lunar ranging was the great improvement in the accuracy of the lunarephemeris[62]and lunar science[67].LLR measures the range from an observatory on the Earth to a retroreflector on the Moon. For the Earth and Moon orbiting the Sun,the scale of relativistic effects is set by the ratio(GM/rc2)≃v2/c2∼10−8.The center-to-center distance of the Moon from the Earth,with mean value385,000km,is variable due to such things as eccentricity,the attraction of the Sun,planets,and the Earth’s bulge,and relativistic corrections.In addition to the lunar orbit,therange from an observatory on the Earth to a retroreflector on the Moon depends on the positionin space of the ranging observatory and the targeted lunar retroreflector.Thus,orientation ofthe rotation axes and the rotation angles of both bodies are important with tidal distortions,plate motion,and relativistic transformations also coming into play.To extract the gravitationalphysics information of interest it is necessary to accurately model a variety of effects[68].For a general review of LLR see[24].A comprehensive paper on tests of gravitationalphysics is[62].A recent test of the EP is in[4]and other GR tests are in[64].An overviewFigure1:Historical accuracy of LLR data from1970to2004.of the LLR gravitational physics tests is given by Nordtvedt[37].Reviews of various tests of relativity,including the contribution by LLR,are given in[58,60].Our recent paper describes the model improvements needed to achieve mm-level accuracy for LLR[66].The most recent LLR results are given in[68].2.3Tests of Relativistic Gravity with LLRLLR offers very accurate laser ranging(weighted rms currently∼2cm or∼5×10−11in frac-tional accuracy)to retroreflectors on the Moon.Analysis of these very precise data contributes to many areas of fundamental and gravitational physics.Thus,these high-precision studies of the Earth-Moon-Sun system provide the most sensitive tests of several key properties of weak-field gravity,including Einstein’s Strong Equivalence Principle(SEP)on which general relativity rests(in fact,LLR is the only current test of the SEP).LLR data yielded the strongest limits to date on variability of the gravitational constant(the way gravity is affected by the expansion of the universe),and the best measurement of the de Sitter precession rate.In this Section we discuss these tests in more details.2.3.1Tests of the Equivalence PrincipleThe Equivalence Principle,the exact correspondence of gravitational and inertial masses,is a central assumption of general relativity and a unique feature of gravitation.EP tests can therefore be viewed in two contexts:tests of the foundations of general relativity,or as searches for new physics.As emphasized by Damour[12,13],almost all extensions to the standard modelof particle physics(with best known extension offered by string theory)generically predict newforces that would show up as apparent violations of the EP.The weak form the EP(the WEP)states that the gravitational properties of strong and electro-weak interactions obey the EP.In this case the relevant test-body differences are their fractional nuclear-binding differences,their neutron-to-proton ratios,their atomic charges,etc. General relativity,as well as other metric theories of gravity,predict that the WEP is exact. However,extensions of the Standard Model of Particle Physics that contain new macroscopic-range quantumfields predict quantum exchange forces that will generically violate the WEP because they couple to generalized‘charges’rather than to mass/energy as does gravity[17,18]. WEP tests can be conducted with laboratory or astronomical bodies,because the relevant differences are in the test-body compositions.Easily the most precise tests of the EP are made by simply comparing the free fall accelerations,a1and a2,of different test bodies.For the case when the self-gravity of the test bodies is negligible and for a uniform external gravityfield, with the bodies at the same distance from the source of the gravity,the expression for the Equivalence Principle takes the most elegant form:∆a= M G M I 2(1)(a1+a2)where M G and M I represent gravitational and inertial masses of each body.The sensitivity of the EP test is determined by the precision of the differential acceleration measurement divided by the degree to which the test bodies differ(position).The strong form of the EP(the SEP)extends the principle to cover the gravitational properties of gravitational energy itself.In other words it is an assumption about the way that gravity begets gravity,i.e.about the non-linear property of gravitation.Although general relativity assumes that the SEP is exact,alternate metric theories of gravity such as those involving scalarfields,and other extensions of gravity theory,typically violate the SEP[30,31, 32,35].For the SEP case,the relevant test body differences are the fractional contributions to their masses by gravitational self-energy.Because of the extreme weakness of gravity,SEP test bodies that differ significantly must have astronomical sizes.Currently the Earth-Moon-Sun system provides the best arena for testing the SEP.The development of the parameterized post-Newtonian formalism[31,56,57],allows one to describe within the common framework the motion of celestial bodies in external gravitational fields within a wide class of metric theories of gravity.Over the last35years,the PPN formalism has become a useful framework for testing the SEP for extended bodies.In that formalism,the ratio of passive gravitational to inertial mass to thefirst order is given by[30,31]:M GMc2 ,(2) whereηis the SEP violation parameter(discussed below),M is the mass of a body and E is its gravitational binding or self-energy:E2Mc2 V B d3x d3yρB(x)ρB(y)EMc2 E=−4.64×10−10andwhere the subscripts E and m denote the Earth and Moon,respectively.The relatively small size bodies used in the laboratory experiments possess a negligible amount of gravitational self-energy and therefore such experiments indicate nothing about the equality of gravitational self-energy contributions to the inertial and passive gravitational masses of the bodies [30].TotesttheSEP onemustutilize planet-sizedextendedbodiesinwhichcase theratioEq.(3)is considerably higher.Dynamics of the three-body Sun-Earth-Moon system in the solar system barycentric inertial frame was used to search for the effect of a possible violation of the Equivalence Principle.In this frame,the quasi-Newtonian acceleration of the Moon (m )with respect to the Earth (E ),a =a m −a E ,is calculated to be:a =−µ∗rM I m µS r SEr 3Sm + M G M I m µS r SEr 3+µS r SEr 3Sm +η E Mc 2 m µS r SEMc 2 E − E n 2−(n −n ′)2n ′2a ′cos[(n −n ′)t +D 0].(8)Here,n denotes the sidereal mean motion of the Moon around the Earth,n ′the sidereal mean motion of the Earth around the Sun,and a ′denotes the radius of the orbit of the Earth around the Sun (assumed circular).The argument D =(n −n ′)t +D 0with near synodic period is the mean longitude of the Moon minus the mean longitude of the Sun and is zero at new Moon.(For a more precise derivation of the lunar range perturbation due to the SEP violation acceleration term in Eq.(6)consult [62].)Any anomalous radial perturbation will be proportional to cos D .Expressed in terms ofη,the radial perturbation in Eq.(8)isδr∼13ηcos D meters [38,21,22].This effect,generalized to all similar three body situations,the“SEP-polarization effect.”LLR investigates the SEP by looking for a displacement of the lunar orbit along the direction to the Sun.The equivalence principle can be split into two parts:the weak equivalence principle tests the sensitivity to composition and the strong equivalence principle checks the dependence on mass.There are laboratory investigations of the weak equivalence principle(at University of Washington)which are about as accurate as LLR[7,1].LLR is the dominant test of the strong equivalence principle.The most accurate test of the SEP violation effect is presently provided by LLR[61,48,23],and also in[24,62,63,4].Recent analysis of LLR data test the EP of∆(M G/M I)EP=(−1.0±1.4)×10−13[68].This result corresponds to a test of the SEP of∆(M G/M I)SEP=(−2.0±2.0)×10−13with the SEP violation parameter η=4β−γ−3found to beη=(4.4±4.5)×10−ing the recent Cassini result for the PPN parameterγ,PPN parameterβis determined at the level ofβ−1=(1.2±1.1)×10−4.2.3.2Other Tests of Gravity with LLRLLR data yielded the strongest limits to date on variability of the gravitational constant(the way gravity is affected by the expansion of the universe),the best measurement of the de Sitter precession rate,and is relied upon to generate accurate astronomical ephemerides.The possibility of a time variation of the gravitational constant,G,wasfirst considered by Dirac in1938on the basis of his large number hypothesis,and later developed by Brans and Dicke in their theory of gravitation(for more details consult[59,60]).Variation might be related to the expansion of the Universe,in which case˙G/G=σH0,where H0is the Hubble constant, andσis a dimensionless parameter whose value depends on both the gravitational constant and the cosmological model considered.Revival of interest in Brans-Dicke-like theories,with a variable G,was partially motivated by the appearance of superstring theories where G is considered to be a dynamical quantity[26].Two limits on a change of G come from LLR and planetary ranging.This is the second most important gravitational physics result that LLR provides.GR does not predict a changing G,but some other theories do,thus testing for this effect is important.The current LLR ˙G/G=(4±9)×10−13yr−1is the most accurate limit published[68].The˙G/G uncertaintyis83times smaller than the inverse age of the universe,t0=13.4Gyr with the value for Hubble constant H0=72km/sec/Mpc from the WMAP data[52].The uncertainty for˙G/G is improving rapidly because its sensitivity depends on the square of the data span.This fact puts LLR,with its more then35years of history,in a clear advantage as opposed to other experiments.LLR has also provided the only accurate determination of the geodetic precession.Ref.[68]reports a test of geodetic precession,which expressed as a relative deviation from GR,is K gp=−0.0019±0.0064.The GP-B satellite should provide improved accuracy over this value, if that mission is successfully completed.LLR also has the capability of determining PPNβandγdirectly from the point-mass orbit perturbations.A future possibility is detection of the solar J2from LLR data combined with the planetary ranging data.Also possible are dark matter tests,looking for any departure from the inverse square law of gravity,and checking for a variation of the speed of light.The accurate LLR data has been able to quickly eliminate several suggested alterations of physical laws.The precisely measured lunar motion is a reality that any proposed laws of attraction and motion must satisfy.The above investigations are important to gravitational physics.The future LLR data will improve the above investigations.Thus,future LLR data of current accuracy would con-tinue to shrink the uncertainty of˙G because of the quadratic dependence on data span.The equivalence principle results would improve more slowly.To make a big improvement in the equivalence principle uncertainty requires improved range accuracy,and that is the motivation for constructing the APOLLO ranging facility in New Mexico.2.4Future LLR Data and APOLLO facilityIt is essential that acquisition of the new LLR data will continue in the future.Accuracies∼2cm are now achieved,and further very useful improvement is expected.Inclusion of improved data into LLR analyses would allow a correspondingly more precise determination of the gravitational physics parameters under study.LLR has remained a viable experiment with fresh results over35years because the data accuracies have improved by an order of magnitude(see Figure1).There are prospects for future LLR station that would provide another order of magnitude improvement.The Apache Point Observatory Lunar Laser-ranging Operation(APOLLO)is a new LLR effort designed to achieve mm range precision and corresponding order-of-magnitude gains in measurements of fundamental physics parameters.For thefirst time in the LLR history,using a3.5m telescope the APOLLO facility will push LLR into a new regime of multiple photon returns with each pulse,enabling millimeter range precision to be achieved[29,66].The anticipated mm-level range accuracy,expected from APOLLO,has a potential to test the EP with a sensitivity approaching10−14.This accuracy would yield sensitivity for parameterβat the level of∼5×10−5and measurements of the relative change in the gravitational constant,˙G/G, would be∼0.1%the inverse age of the universe.The overwhelming advantage APOLLO has over current LLR operations is a3.5m astro-nomical quality telescope at a good site.The site in southern New Mexico offers high altitude (2780m)and very good atmospheric“seeing”and image quality,with a median image resolu-tion of1.1arcseconds.Both the image sharpness and large aperture conspire to deliver more photons onto the lunar retroreflector and receive more of the photons returning from the re-flectors,pared to current operations that receive,on average,fewer than0.01 photons per pulse,APOLLO should be well into the multi-photon regime,with perhaps5–10 return photons per pulse.With this signal rate,APOLLO will be efficient atfinding and track-ing the lunar return,yielding hundreds of times more photons in an observation than current√operations deliver.In addition to the significant reduction in statistical error(useful).These new reflectors on the Moon(and later on Mars)can offer significant navigational accuracy for many space vehicles on their approach to the lunar surface or during theirflight around the Moon,but they also will contribute significantly to fundamental physics research.The future of lunar ranging might take two forms,namely passive retroreflectors and active transponders.The advantages of new installations of passive retroreflector arrays are their long life and simplicity.The disadvantages are the weak returned signal and the spread of the reflected pulse arising from lunar librations(apparent changes in orientation of up to10 degrees).Insofar as the photon timing error budget is dominated by the libration-induced pulse spread—as is the case in modern lunar ranging—the laser and timing system parameters do√not influence the net measurement uncertainty,which simply scales as1/3Laser Ranging to MarsThere are three different experiments that can be done with accurate ranges to Mars:a test of the SEP(similar to LLR),a solar conjunction experiment measuring the deflection of light in the solar gravity,similar to the Cassini experiment,and a search for temporal variation in the gravitational constant G.The Earth-Mars-Sun-Jupiter system allows for a sensitive test of the SEP which is qualitatively different from that provided by LLR[3].Furthermore,the outcome of these ranging experiments has the potential to improve the values of the two relativistic parameters—a combination of PPN parametersη(via test of SEP)and a direct observation of the PPN parameterγ(via Shapiro time delay or solar conjunction experiments).(This is quite different compared to LLR,as the small variation of Shapiro time delay prohibits very accurate independent determination of the parameterγ).The Earth-Mars range would also provide for a very accurate test of˙G/G.This section qualitatively addresses the near-term possibility of laser ranging to Mars and addresses the above three effects.3.1Planetary Test of the SEP with Ranging to MarsEarth-Mars ranging data can provide a useful estimate of the SEP parameterηgiven by Eq.(7). It was demonstrated in[3]that if future Mars missions provide ranging measurements with an accuracy ofσcentimeters,after ten years of ranging the expected accuracy for the SEP parameterηmay be of orderσ×10−6.These ranging measurements will also provide the most accurate determination of the mass of Jupiter,independent of the SEP effect test.It has been observed previously that a measurement of the Sun’s gravitational to inertial mass ratio can be performed using the Sun-Jupiter-Mars or Sun-Jupiter-Earth system[33,47,3]. The question we would like to answer here is how accurately can we do the SEP test given the accurate ranging to Mars?We emphasize that the Sun-Mars-Earth-Jupiter system,though governed basically by the same equations of motion as Sun-Earth-Moon system,is significantly different physically.For a given value of SEP parameterηthe polarization effects on the Earth and Mars orbits are almost two orders of magnitude larger than on the lunar orbit.Below we examine the SEP effect on the Earth-Mars range,which has been measured as part of the Mariner9and Viking missions with ranging accuracy∼7m[48,44,41,43].The main motivation for our analysis is the near-future Mars missions that should yield ranging data, accurate to∼1cm.This accuracy would bring additional capabilities for the precision tests of fundamental and gravitational physics.3.1.1Analytical Background for a Planetary SEP TestThe dynamics of the four-body Sun-Mars-Earth-Jupiter system in the Solar system barycentric inertial frame were considered.The quasi-Newtonian acceleration of the Earth(E)with respect to the Sun(S),a SE=a E−a S,is straightforwardly calculated to be:a SE=−µ∗SE·r SE MI Eb=M,Jµb r bS r3bE + M G M I E b=M,Jµb r bS。
State Space Reconstruction for Multivariate Time Series Prediction
a r X i v :0809.2220v 1 [n l i n .C D ] 12 S e p 2008APS/123-QEDState Space Reconstruction for Multivariate Time Series PredictionI.Vlachos ∗and D.Kugiumtzis †Department of Mathematical,Physical and Computational Sciences,Faculty of Technology,Aristotle University of Thessaloniki,Greece(Dated:September 12,2008)In the nonlinear prediction of scalar time series,the common practice is to reconstruct the state space using time-delay embedding and apply a local model on neighborhoods of the reconstructed space.The method of false nearest neighbors is often used to estimate the embedding dimension.For prediction purposes,the optimal embedding dimension can also be estimated by some prediction error minimization criterion.We investigate the proper state space reconstruction for multivariate time series and modify the two abovementioned criteria to search for optimal embedding in the set of the variables and their delays.We pinpoint the problems that can arise in each case and compare the state space reconstructions (suggested by each of the two methods)on the predictive ability of the local model that uses each of them.Results obtained from Monte Carlo simulations on known chaotic maps revealed the non-uniqueness of optimum reconstruction in the multivariate case and showed that prediction criteria perform better when the task is prediction.PACS numbers:05.45.Tp,02.50.Sk,05.45.aKeywords:nonlinear analysis,multivariate analysis,time series,local prediction,state space reconstructionI.INTRODUCTIONSince its publication Takens’Embedding Theorem [1](and its extension,the Fractal Delay Embedding Preva-lence Theorem by Sauer et al.[2])has been used in time series analysis in many different settings ranging from system characterization and approximation of invariant quantities,such as correlation dimension and Lyapunov exponents,to prediction and noise-filtering [3].The Em-bedding Theorem implies that although the true dynam-ics of a system may not be known,equivalent dynamics can be obtained under suitable conditions using time de-lays of a single time series,treated as an one-dimensional projection of the system trajectory.Most applications of the Embedding Theorem deal with univariate time series,but often measurements of more than one quantities related to the same dynamical system are available.One of the first uses of multivari-ate embedding was in the context of spatially extended systems where embedding vectors were constructed from data representing the same quantity measured simulta-neously at different locations [4,5].Multivariate em-bedding was used for noise reduction [6]and for surro-gate data generation with equal individual delay times and equal embedding dimensions for each time series [7].In nonlinear multivariate prediction,the prediction with local models on a space reconstructed from a different time series of the same system was studied in [8].This study was extended in [9]by having the reconstruction utilize all of the observed time series.Multivariate em-bedding with the use of independent components analysis was considered in [10]and more recently multivariate em-2as x n=h(y n).Despite the apparent loss of information of the system dynamics by the projection,the system dynamics may be recovered through suitable state space reconstruction from the scalar time series.A.Reconstruction of the state space According to Taken’s embedding theorem a trajectory formed by the points x n of time-delayed components from the time series{x n}N n=1asx n=(x n−(m−1)τ,x n−(m−2)τ,...,x n),(1)under certain genericity assumptions,is an one-to-one mapping of the original trajectory of y n provided that m is large enough.Given that the dynamical system“lives”on an attrac-tor A⊂Γ,the reconstructed attractor˜A through the use of the time-delay vectors is topologically equivalent to A.A sufficient condition for an appropriate unfolding of the attractor is m≥2d+1where d is the box-counting dimension of A.The embedding process is visualized in the following graphy n∈A⊂ΓF→y n+1∈A⊂Γ↓h↓hx n∈R x n+1∈R↓e↓ex n∈˜A⊂R m G→x n+1∈˜A⊂R mwhere e is the embedding procedure creating the delay vectors from the time series and G is the reconstructed dynamical system on˜A.G preserves properties of the unknown F on the unknown attractor A that do not change under smooth coordinate transformations.B.Univariate local predictionFor a given state space reconstruction,the local predic-tion at a target point x n is made with a model estimated on the K nearest neighboring points to x n.The local model can have a simple form,such as the zeroth order model(the average of the images of the nearest neigh-bors),but here we consider the linear modelˆx n+1=a(n)x n+b(n),where the superscript(n)denotes the dependence of the model parameters(a(n)and b(n))on the neighborhood of x n.The neighborhood at each target point is defined either by afixed number K of nearest neighbors or by a distance determining the borders of the neighborhood giving a varying K with x n.C.Selection of embedding parametersThe two parameters of the delay embedding in(1)are the embedding dimension m,i.e.the number of compo-nents in x n and the delay timeτ.We skip the discussion on the selection ofτas it is typically set to1in the case of discrete systems that we focus on.Among the ap-proaches for the selection of m we choose the most popu-lar method of false nearest neighbors(FNN)and present it briefly below[13].The measurement function h projects distant points {y n}of the original attractor to close values of{x n}.A small m may still give badly projected points and we seek the reconstructed state space of the smallest embed-ding dimension m that unfolds the attractor.This idea is implemented as follows.For each point x m n in the m-dimensional reconstructed state space,the distance from its nearest neighbor x mn(1)is calculated,d(x m n,x mn(1))=x m n−x mn(1).The dimension of the reconstructed state space is augmented by1and the new distance of thesevectors is calculated,d(x m+1n,x m+1n(1))= x m+1n−x m+1n(1). If the ratio of the two distances exceeds a predefined tol-erance threshold r the two neighbors are classified as false neighbors,i.e.r n(m)=d(x m+1n,x m+1n(1))3 III.MULTIV ARIATE EMBEDDINGIn Section II we gave a summary of the reconstructiontechnique for a deterministic dynamical system from ascalar time series generated by the system.However,it ispossible that more than one time series are observed thatare possibly related to the system under investigation.For p time series measured simultaneously from the samedynamical system,a measurement function H:Γ→R pis decomposed to h i,i=1,...,p,defined as in Section II,giving each a time series{x i,n}N n=1.According to the dis-cussion on univariate embedding any of the p time seriescan be used for reconstruction of the system dynamics,or better,the most suitable time series could be selectedafter proper investigation.In a different approach all theavailable time series are considered and the analysis ofthe univariate time series is adjusted to the multivariatetime series.A.From univariate to multivariate embeddingGiven that there are p time series{x i,n}N n=1,i=1,...,p,the equivalent to the reconstructed state vec-tor in(1)for the case of multivariate embedding is of theformx n=(x1,n−(m1−1)τ1,x1,n−(m1−2)τ1,...,x1,n,x2,n−(m2−1)τ2,...,x2,n,...,x p,n)(3)and are defined by an embedding dimension vector m= (m1,...,m p)that indicates the number of components used from each time series and a time delay vector τ=(τ1,...,τp)that gives the delays for each time series. The corresponding graph for the multivariate embedding process is shown below.y n∈A⊂ΓF→y n+1∈A⊂Γւh1↓h2...ցhpւh1↓h2...ցhpx1,n x2,n...x p,n x1,n+1x2,n+1...x p,n+1ցe↓e...ւeցe↓e...ւex n∈˜A⊂R M G→x n+1∈˜A⊂R MThe total embedding dimension M is the sum of the individual embedding dimensions for each time seriesM= p i=1m i.Note that if redundant or irrelevant information is present in the p time series,only a sub-set of them may be represented in the optimal recon-structed points x n.The selection of m andτfollows the same principles as for the univariate case:the attrac-tor should be fully unfolded and the components of the embedding vectors should be uncorrelated.A simple se-lection rule suggests that all individual delay times and embedding dimensions are the same,i.e.m=m1and τ=τ1with1a p-vector of ones[6,7].Here,we set againτi=1,i=1,...,p,but we consider bothfixed and varying m i in the implementation of the FNN method (see Section III D).B.Multivariate local predictionThe prediction for each time series x i,n,i=1,...,p,is performed separately by p local models,estimated as in the case of univariate time series,but for reconstructed points formed potentially from all p time series as given in(3)(e.g.see[9]).We propose an extension of the NRMSE for the pre-diction of one time series to account for the error vec-tors comprised of the individual prediction errors for each of the predicted time series.If we have one step ahead predictions for the p available time series,i.e.ˆx i,n, i=1,...,p(for a range of current times n−1),we define the multivariate NRMSENRMSE=n (x1,n−¯x1,...,x p,n−¯x p) 2(4)where¯x i is the mean of the actual values of x i,n over all target times n.C.Problems and restrictions of multivariatereconstructionsA major problem in the multivariate case is the prob-lem of identification.There are often not unique m and τembedding parameters that unfold fully the attractor.A trivial example is the Henon map[17]x n+1=1.4−x2n+y ny n+1=0.3x n(5) It is known that for the state space reconstruction from the observable x n the appropriate embedding parame-ters are m=2andτ=1.Due to the fact that y n is a lagged multiple of x n the attractor can obviously be reconstructed from the bivariate time series{x n,y n} equally well with any of the following two-dimensional embedding schemesx n=(x n,x n−1)x n=(x n,y n)x n=(y n,y n−1) since they are essentially the same.This example shows also the problem of redundant information,e.g.the state space reconstruction would not improve by augmenting the delay vector x n=(x n,x n−1)with the component y n that actually duplicates x n−1.Redundancy is inevitable in multivariate time series as synchronous observations of the different time series are generally correlated and the fact that these observations are used as components in the same embedding vector adds redundant information in them.We note here that in the case of continuous dynamical systems,the delay parameterτi may be se-lected so that the components of the i time series are not correlated with each other,but this does not imply that they are not correlated to components from another time series.4 A different problem is that of irrelevance,whenseries that are not generated by the same dynamicaltem are included in the reconstruction procedure.may be the case even when a time series is connectedtime series generated by the system underAn issue of concern is also the fact thatdata don’t always have the same data ranges andtances calculated on delay vectors withdifferent ranges may depend highly on only some ofcomponents.So it is often preferred to scale all theto have either the same variance or be in the samerange.For our study we choose to scale the data torange[0,1].D.Selection of the embedding dimension vector Taking into account the problems in the state space reconstruction from multivariate time series,we present three methods for determining m,two based on the false nearest neighbor algorithm,which we name FNN1and FNN2,and one based on local models which we call pre-diction error minimization criterion(PEM).The main idea of the FNN algorithms is as for the univariate case.Starting from a small value the embed-ding dimension is increased by including delay compo-nents from the p time series and the percentage of the false nearest neighbors is calculated until it falls to the zero level.The difference of the two FNN methods is on the way that m is increased.For FNN1we restrict the state space reconstruction to use the same embedding dimension for each of the p time series,i.e.m=(m,m,...,m)for a given m.To assess whether m is sufficient,we consider all delay embeddings derived by augmenting the state vector of embedding di-mension vector(m,m,...,m)with a single delayed vari-able from any of the p time series.Thus the check for false nearest neighbors in(2)yields the increase from the embedding dimension vector(m,m,...,m)to each of the embedding dimension vectors(m+1,m,...,m), (m,m+1,...,m),...,(m,m,...,m+1).Then the algo-rithm stops at the optimal m=(m,m,...,m)if the zero level percentage of false nearest neighbors is obtained for all p cases.A sketch of thefirst two steps for a bivariate time series is shown in Figure1(a).This method has been commonly used in multivariate reconstruction and is more appropriate for spatiotem-porally distributed data(e.g.see the software package TISEAN[18]).A potential drawback of FNN1is that the selected total embedding dimension M is always a multiple of p,possibly introducing redundant informa-tion in the embedding vectors.We modify the algorithm of FNN1to account for any form of the embedding dimension vector m and the total embedding dimension M is increased by one at each step of the algorithm.Let us suppose that the algorithm has reached at some step the total embedding dimension M. For this M all the combinations of the components of the embedding dimension vector m=(m1,m2,...,m p)are considered under the condition M= p i=1m i.Then for each such m=(m1,m2,...,m p)all the possible augmen-tations with one dimension are checked for false nearest neighbors,i.e.(m1+1,m2,...,m p),(m1,m2+1,...,m p), ...,(m1,m2,...,m p+1).A sketch of thefirst two steps of the extended FNN algorithm,denoted as FNN2,for a bivariate time series is shown in Figure1(b).The termination criterion is the drop of the percent-age of false nearest neighbors to the zero level at every increase of M by one for at least one embedding dimen-sion vector(m1,m2,...,m p).If more than one embedding dimension vectors fulfill this criterion,the one with the smallest cumulative FNN percentage is selected,where the cumulative FNN percentage is the sum of the p FNN percentages for the increase by one of the respective com-ponent of the embedding dimension vector.The PEM criterion for the selection of m= (m1,m2,...,m p)is simply the extension of the goodness-of-fit or prediction criterion in the univariate case to account for the multiple ways the delay vector can be formed from the multivariate time series.Thus for all possible p-plets of(m1,m2,...,m p)from(1,0,...,0), (0,1,...,0),etc up to some vector of maximum embed-ding dimensions(m max,m max,...,m max),the respective reconstructed state spaces are created,local linear mod-els are applied and out-of-sample prediction errors are computed.So,totally p m max−1embedding dimension vectors are compared and the optimal is the one that gives the smallest multivariate NRMSE as defined in(4).IV.MONTE CARLO SIMULATIONS ANDRESULTSA.Monte Carlo setupWe test the three methods by performing Monte Carlo simulations on a variety of known nonlinear dynamical systems.The embedding dimension vectors are selected using the three methods on100different realizations of each system and the most frequently selected embedding dimension vectors for each method are tracked.Also,for each realization and selected embedding dimension vec-5ate NRMSE over the100realizations for each method is then used as an indicator of the performance of each method in prediction.The selection of the embedding dimension vector by FNN1,FNN2and PEM is done on thefirst three quarters of the data,N1=3N/4,and the multivariate NRMSE is computed on the last quarter of the data(N−N1).For PEM,the same split is used on the N1data,so that N2= 3N1/4data are used tofind the neighbors(training set) and the rest N1−N2are used to compute the multivariate NRMSE(test set)and decide for the optimal embedding dimension vector.A sketch of the split of the data is shown in Figure2.The number of neighbors for the local models in PEM varies with N and we set K N=10,25,50 for time series lengths N=512,2048,8192,respectively. The parameters of the local linear model are estimated by ordinary least squares.For all methods the investigation is restricted to m max=5.The multivariate time series are derived from nonlin-ear maps of varying dimension and complexity as well as spatially extended maps.The results are given below for each system.B.One and two Ikeda mapsThe Ikeda map is an example of a discrete low-dimensional chaotic system in two variables(x n,y n)de-fined by the equations[19]z n+1=1+0.9exp(0.4i−6i/(1+|z n|2)),x n=Re(z n),y n=Im(z n),where Re and Im denote the real and imaginary part,re-spectively,of the complex variable z n.Given the bivari-ate time series of(x n,y n),both FNN methods identify the original vector x n=(x n,y n)andfind m=(1,1)as optimal at all realizations,as shown in Table I.On the other hand,the PEM criterionfinds over-embedding as optimal,but this improves slightly the pre-diction,which as expected improves with the increase of N.Next we consider the sum of two Ikeda maps as a more complex and higher dimensional system.The bivariateI:Dimension vectors and NRMSE for the Ikeda map.2,3and4contain the embedding dimension vectorsby their respective frequency of occurrenceNRMSEFNN1PEM FNN2 512(1,1)1000.0510.032 (1,1)100(2,2)1000.028 8192(1,1)1000.0130.003II:Dimension vectors and NRMSE for the sum ofmapsNRMSEFNN1PEM FNN2 512(2,2)650.4560.447(1,3)26(3,3)95(2,3)540.365(2,2)3(2,2)448192(2,3)430.2600.251(1,4)37time series are generated asx n=Re(z1,n+z2,n),y n=Im(z1,n+z2,n).The results of the Monte Carlo simulations shown in Ta-ble II suggest that the prediction worsens dramatically from that in Table I and the total embedding dimension M increases with N.The FNN2criterion generally gives multiple optimal m structures across realizations and PEM does the same but only for small N.This indicates that high complex-ity degrades the performance of the algorithms for small sample sizes.PEM is again best for predictions but over-all we do not observe large differences in the three meth-ods.An interesting observation is that although FNN2finds two optimal m with high frequencies they both give the same M.This reflects the problem of identification, where different m unfold the attractor equally well.This feature cannot be observed in FNN1because the FNN1 algorithm inspects fewer possible vectors and only one for each M,where M can only be multiple of p(in this case(1,1)for M=2,(2,2)for M=4,etc).On the other hand,PEM criterion seems to converge to a single m for large N,which means that for the sum of the two Ikeda maps this particular structure gives best prediction re-sults.Note that there is no reason that the embedding dimension vectors derived from FNN2and PEM should match as they are selected under different conditions. Moreover,it is expected that the m selected by PEM gives always the lowest average of multivariate NRMSE as it is selected to optimize prediction.TABLE III:Dimension vectors and NRMSE for the KDR mapNRMSE FNN1PEM FNN2512(0,0,2,2)30(1,1,1,1)160.7760.629 (1,1,1,1)55(2,2,2,2)39(0,2,1,1)79(0,1,0,1)130.6598192(2,1,1,1)40(1,1,1,1)140.5580.373TABLE IV:Dimension vectors and NRMSE for system of Driver-Response Henon systemEmbedding dimensionsN FNN1PEM FNN2512(2,2)100(2,2)75(2,1)100.196(2,2)100(3,2)33(2,2)250.127(2,2)100(3,0)31(0,3)270.0122048(2,2)100(2,2)1000.093(2,2)100(3,3)45(4,3)450.084(2,2)100(0,3)20(3,0)190.0068192(2,2)100(2,2)1000.051(2,2)100(3,3)72(4,3)250.027(2,2)100(0,4)31(4,0)300.002TABLE V:Dimension vectors and NRMSE for Lattice of3coupled Henon mapsEmbedding dimensionsN FNN1PEM FNN2512(2,2,2)94(1,1,1)6(1,2,1)29(1,1,2)230.298(2,2,2)98(1,1,1)2(2,0,2)44(2,1,1)220.2282048(2,2,2)100(1,2,2)34(2,2,1)300.203(2,2,2)100(2,1,2)48(2,0,2)410.1318192(2,2,2)100(2,2,2)97(3,2,3)30.174(2,2,2)100(2,1,2)79(3,2,3)190.084NRMSEC FNN2FNN1PEM0.4(1,1,1,1)42(1,0,2,1)170.2850.2880.8(1,1,1,1)40(1,0,1,2)170.3140.2910.4(1,1,1,1)88(1,1,1,2)70.2290.1900.8(1,1,1,1)36(1,0,2,1)330.2250.1630.4(1,1,1,1)85(1,2,1,1)80.1970.1370.8(1,2,0,1)31(1,0,2,1)220.1310.072 PEM cannot distinguish the two time series and selectswith almost equal frequencies vectors of the form(m,0)and(0,m)giving again over-embedding as N increases.Thus PEM does not reveal the coupling structure of theunderlying system and picks any embedding dimensionstructure among a range of structures that give essen-tially equivalent predictions.Here FNN2seems to de-tect sufficiently the underlying coupling structure in thesystem resulting in a smaller total embedding dimensionthat gives however the same level of prediction as thelarger M suggested by FNN1and slightly smaller thanthe even larger M found by PEM.ttices of coupled Henon mapsThe last system is an example of spatiotemporal chaosand is defined as a lattice of k coupled Henon maps{x i,n,y i,n}k i=1[22]specified by the equationsx i,n+1=1.4−((1−C)x i,n+C(x i−1,n+x i+1,n)ple size,at least for the sizes we used in the simulations. Such a feature shows lack of consistency of the PEM cri-terion and suggests that the selection is led from factors inherent in the prediction process rather than the quality of the reconstructed attractor.For example the increase of embedding dimension with the sample size can be ex-plained by the fact that more data lead to abundance of close neighbors used in local prediction models and this in turn suggests that augmenting the embedding vectors would allow to locate the K neighbors used in the model. On the other hand,the two schemes used here that ex-tend the method of false nearest neighbors(FNN)to mul-tivariate time series aim atfinding minimum embedding that unfolds the attractor,but often a higher embedding gives better prediction results.In particular,the sec-ond scheme(FNN2)that explores all possible embedding structures gives consistent selection of an embedding of smaller dimension than that selected by PEM.Moreover, this embedding could be justified by the underlying dy-namics of the known systems we tested.However,lack of consistency of the selected embedding was observed with all methods for small sample sizes(somehow expected due to large variance of any estimate)and for the cou-pled maps(probably due to the presence of more than one optimal embeddings).In this work,we used only a prediction performance criterion to assess the quality of state space reconstruc-tion,mainly because it has the most practical relevance. There is no reason to expect that PEM would be found best if the assessment was done using another criterion not based on prediction.However,the reference(true)value of other measures,such as the correlation dimen-sion,are not known for all systems used in this study.An-other constraint of this work is that only noise-free multi-variate time series from discrete systems are encountered, so that the delay parameter is not involved in the state space reconstruction and the effect of noise is not studied. It is expected that the addition of noise would perplex further the process of selecting optimal embedding di-mension and degrade the performance of the algorithms. For example,we found that in the case of the Henon map the addition of noise of equal magnitude to the two time series of the system makes the criteria to select any of the three equivalent embeddings((2,0),(0,2),(1,1))at random.It is in the purpose of the authors to extent this work and include noisy multivariate time series,also fromflows,and search for other measures to assess the performance of the embedding selection methods.AcknowledgmentsThis paper is part of the03ED748research project,im-plemented within the framework of the”Reinforcement Programme of Human Research Manpower”(PENED) and co-financed at90%by National and Community Funds(25%from the Greek Ministry of Development-General Secretariat of Research and Technology and75% from E.U.-European Social Fund)and at10%by Rik-shospitalet,Norway.[1]F.Takens,Lecture Notes in Mathematics898,365(1981).[2]T.Sauer,J.A.Yorke,and M.Casdagli,Journal of Sta-tistical Physics65,579(1991).[3]H.Kantz and T.Schreiber,Nonlinear Time Series Anal-ysis(Cambridge University Press,1997).[4]J.Guckenheimer and G.Buzyna,Physical Review Let-ters51,1438(1983).[5]M.Paluˇs,I.Dvoˇr ak,and I.David,Physica A StatisticalMechanics and its Applications185,433(1992).[6]R.Hegger and T.Schreiber,Physics Letters A170,305(1992).[7]D.Prichard and J.Theiler,Physical Review Letters73,951(1994).[8]H.D.I.Abarbanel,T.A.Carroll,,L.M.Pecora,J.J.Sidorowich,and L.S.Tsimring,Physical Review E49, 1840(1994).[9]L.Cao,A.Mees,and K.Judd,Physica D121,75(1998),ISSN0167-2789.[10]J.P.Barnard,C.Aldrich,and M.Gerber,Physical Re-view E64,046201(2001).[11]S.P.Garcia and J.S.Almeida,Physical Review E(Sta-tistical,Nonlinear,and Soft Matter Physics)72,027205 (2005).[12]Y.Hirata,H.Suzuki,and K.Aihara,Physical ReviewE(Statistical,Nonlinear,and Soft Matter Physics)74, 026202(2006).[13]M.B.Kennel,R.Brown,and H.D.I.Abarbanel,Phys-ical Review A45,3403(1992).[14]D.T.Kaplan,in Chaos in Communications,edited byL.M.Pecora(SPIE-The International Society for Optical Engineering,Bellingham,Washington,98227-0010,USA, 1993),pp.236–240.[15]B.Chun-Hua and N.Xin-Bao,Chinese Physics13,633(2004).[16]R.Hegger and H.Kantz,Physical Review E60,4970(1999).[17]M.H´e non,Communications in Mathematical Physics50,69(1976).[18]R.Hegger,H.Kantz,and T.Schreiber,Chaos:An Inter-disciplinary Journal of Nonlinear Science9,413(1999).[19]K.Ikeda,Optics Communications30,257(1979).[20]C.Grebogi,E.Kostelich,E.O.Ott,and J.A.Yorke,Physica D25(1987).[21]S.J.Schiff,P.So,T.Chang,R.E.Burke,and T.Sauer,Physical Review E54,6708(1996).[22]A.Politi and A.Torcini,Chaos:An InterdisciplinaryJournal of Nonlinear Science2,293(1992).。
上外考研翻译硕士英语天文学专业词汇整理分享
上外考研翻译硕士英语天文学专业词汇整理分享find 发见陨星finder chart 证认图finderscope 寻星镜first-ascent giant branch初升巨星支first giant branch 初升巨星支flare puff 耀斑喷焰flat field 平场flat field correction 平场改正flat fielding 平场处理flat-spectrum radio quasar 平谱射电类星体flux standard 流量标准星flux-tube dynamics 磁流管动力学f-mode f 模、基本模following limb 东边缘、后随边缘foreground galaxy 前景星系foreground galaxy cluster 前景星系团formal accuracy 形式精度Foucaultgram 傅科检验图样Foucault knife-edge test 傅科刀口检验fourth cosmic velocity 第四宇宙速度frame transfer 帧转移Fresnel lens 菲涅尔透镜fuzz 展云Galactic aggregate 银河星集Galactic astronomy 银河系天文Galactic bar 银河系棒galactic bar 星系棒galactic cannibalism 星系吞食galactic content 星系成分galactic merge 星系并合galactic pericentre 近银心点Galactocentric distance 银心距galaxy cluster 星系团Galle ring 伽勒环Galilean transformation 伽利略变换Galileo 〈伽利略〉木星探测器gas-dust complex 气尘复合体Genesis rock 创世岩Gemini Telescope 大型双子望远镜giant granulation 巨米粒组织giant granule 巨米粒giant radio pulse 巨射电脉冲Ginga 〈星系〉X 射线天文卫星Giotto 〈乔托〉空间探测器glassceramic 微晶玻璃glitch activity 自转突变活动global change 全球变化global sensitivity 全局灵敏度GMC, giant molecular cloud 巨分子云g-mode g 模、重力模gold spot 金斑病GONG, Global Oscillation Network 太阳全球振荡监测网GPS, global positioning system 全球定位系统Granat 〈石榴〉号天文卫星grand design spiral 宏象旋涡星系gravitational astronomy 引力天文gravitational lensing 引力透镜效应gravitational micro-lensing 微引力透镜效应great attractor 巨引源Great Dark Spot 大暗斑Great White Spot 大白斑grism 棱栅GRO, Gamma-Ray Observatory γ射线天文台guidscope 导星镜GW Virginis star 室女GW 型星habitable planet 可居住行星Hakucho 〈天鹅〉X 射线天文卫星Hale Telescope 海尔望远镜halo dwarf 晕族矮星halo globular cluster 晕族球状星团Hanle effect 汉勒效应hard X-ray source 硬X 射线源Hay spot 哈伊斑HEAO, High-Energy Astronomical 〈HEAO〉高能天文台Observatory heavy-element star 重元素星heiligenschein 灵光Helene 土卫十二helicity 螺度heliocentric radial velocity 日心视向速度heliomagnetosphere 日球磁层helioseismology 日震学helium abundance 氦丰度helium main-sequence 氦主序helium-strong star 强氦线星helium white dwarf 氦白矮星Helix galaxy ( NGC 2685 ) 螺旋星系Herbig Ae star 赫比格Ae 型星Herbig Be star 赫比格Be 型星Herbig-Haro flow 赫比格-阿罗流Herbig-Haro shock wave 赫比格-阿罗激波hidden magnetic flux 隐磁流high-field pulsar 强磁场脉冲星highly polarized quasar ( HPQ ) 高偏振类星体high-mass X-ray binary 大质量X 射线双星high-metallicity cluster 高金属度星团;high-resolution spectrograph 高分辨摄谱仪high-resolution spectroscopy 高分辨分光high - z 大红移Hinotori 〈火鸟〉太阳探测器Hipparcos, High Precision Parallax 〈依巴谷〉卫星Collecting SatelliteHipparcos and Tycho Catalogues 〈依巴谷〉和〈第谷〉星表holographic grating 全息光栅Hooker Telescope 胡克望远镜host galaxy 寄主星系hot R Coronae Borealis star 高温北冕R 型星HST, Hubble Space Telescope 哈勃空间望远镜Hubble age 哈勃年龄Hubble distance 哈勃距离Hubble parameter 哈勃参数Hubble velocity 哈勃速度hump cepheid 驼峰造父变星Hyad 毕团星hybrid-chromosphere star 混合色球星hybrid star 混合大气星hydrogen-deficient star 缺氢星hydrogenous atmosphere 氢型大气hypergiant 特超巨星Ida 艾达( 小行星243号)IEH, International Extreme Ultraviolet Hitchhiker〈IEH〉国际极紫外飞行器IERS, International Earth Rotation Service国际地球自转服务image deconvolution 图象消旋image degradation 星象劣化image dissector 析象管image distoration 星象复原image photon counting system 成象光子计数系统image sharpening 星象增锐image spread 星象扩散度imaging polarimetry 成象偏振测量imaging spectrophotometry 成象分光光度测量immersed echelle 浸渍阶梯光栅impulsive solar flare 脉冲太阳耀斑infralateral arc 外侧晕弧infrared CCD 红外CCDinfrared corona 红外冕infrared helioseismology 红外日震学infrared index 红外infrared observatory 红外天文台infrared spectroscopy 红外分光initial earth 初始地球initial mass distribution 初始质量分布initial planet 初始行星initial star 初始恒星initial sun 初始太阳inner coma 内彗发inner halo cluster 内晕族星团integrability 可积性Integral Sign galaxy ( UGC 3697 ) 积分号星系integrated diode array ( IDA ) 集成二极管阵intensified CCD 增强CCD Intercosmos 〈国际宇宙〉天文卫星interline transfer 行间转移intermediate parent body 中间母体intermediate polar 中介偏振星international atomic time 国际原子时International Celestial Reference 国际天球参考系Frame ( ICRF ) intraday variation 快速变化intranetwork element 网内元intrinsic dispersion 内廪弥散度ion spot 离子斑IPCS, Image Photon Counting System 图象光子计数器IRIS, Infrared Imager / Spectrograph 红外成象器/摄谱仪IRPS, Infrared Photometer / Spectro- meter 红外光度计/分光计irregular cluster 不规则星团; 不规则星系团IRTF, NASA Infrared Telescope 〈IRTF〉美国宇航局红外Facility 望远镜IRTS, Infrared Telescope in Space 〈IRTS〉空间红外望远镜ISO, Infrared Space Observatory 〈ISO〉红外空间天文台isochrone method 等龄线法IUE, International Ultraviolet Explorer〈IUE〉国际紫外探测器Jewel Box ( NGC 4755 ) 宝盒星团Jovian magnetosphere 木星磁层Jovian ring 木星环Jovian ringlet 木星细环Jovian seismology 木震学jovicentric orbit 木心轨道J-type star J 型星Juliet 天卫十一Jupiter-crossing asteroid 越木小行星Kalman filter 卡尔曼滤波器KAO, Kuiper Air-borne Observatory 〈柯伊伯〉机载望远镜Keck ⅠTelescope 凯克Ⅰ望远镜Keck ⅡTelescope 凯克Ⅱ望远镜Kuiper belt 柯伊伯带Kuiper disk 柯伊伯盘LAMOST, Large Multi-Object Fibre Spectroscopic Telescope大型多天体分光望远镜Laplacian plane 拉普拉斯平面late cluster 晚型星系团LBT, Large Binocular Telescope 〈LBT〉大型双筒望远镜lead oxide vidicon 氧化铅光导摄象管Leo Triplet 狮子三重星系LEST, Large Earth-based Solar Telescope〈LEST〉大型地基太阳望远镜level-Ⅰcivilization Ⅰ级文明level-Ⅱcivilization Ⅱ级文明level-Ⅲcivilization Ⅲ级文明Leverrier ring 勒威耶环Liapunov characteristic number 李雅普诺夫特征数light crown 轻冕玻璃light echo 回光light-gathering aperture 聚光孔径light pollution 光污染light sensation 光感line image sensor 线成象敏感器line locking 线锁line-ratio method 谱线比法Liner, low ionization nuclear 低电离核区emission-line regionline spread function 线扩散函数LMT, Large Millimeter Telescope 〈LMT〉大型毫米波望远镜local galaxy 局域星系local inertial frame 局域惯性架local inertial system 局域惯性系local object 局域天体local star 局域恒星look-up table ( LUT ) 对照表low-mass X-ray binary 小质量X 射线双星low-metallicity cluster 低金属度星团;low-resolution spectrograph 低分辨摄谱仪low-resolution spectroscopy 低分辨分光low - z 小红移luminosity mass 光度质量luminosity segregation 光度层化luminous blue variable 高光度蓝变星lunar atmosphere 月球大气lunar chiaroscuro 月相图Lunar Prospector 〈月球勘探者〉Ly-α forest 莱曼-α森林MACHO ( massive compact halo object ) 晕族大质量致密天体Magellan 〈麦哲伦〉金星探测器Magellan Telescope 〈麦哲伦〉望远镜magnetic canopy 磁蓬magnetic cataclysmic variable 磁激变变星magnetic curve 磁变曲线magnetic obliquity 磁夹角magnetic period 磁变周期magnetic phase 磁变相位magnitude range 星等范围main asteroid belt 主小行星带main-belt asteroid 主带小行星main resonance 主共振main-sequence band 主序带Mars-crossing asteroid 越火小行星Mars Pathfinder 火星探路者mass loss rate 质量损失率mass segregation 质量层化Mayall Telescope 梅奥尔望远镜Mclntosh classification 麦金托什分类McMullan camera 麦克马伦电子照相机mean motion resonance 平均运动共振membership of cluster of galaxies 星系团成员membership of star cluster 星团成员merge 并合merger 并合星系; 并合恒星merging galaxy 并合星系merging star 并合恒星mesogranulation 中米粒组织mesogranule 中米粒metallicity 金属度metallicity gradient 金属度梯度metal-poor cluster 贫金属星团metal-rich cluster 富金属星团MGS, Mars Global Surveyor 火星环球勘测者micro-arcsec astrometry 微角秒天体测量microchannel electron multiplier 微通道电子倍增管microflare 微耀斑microgravitational lens 微引力透镜microgravitational lensing 微引力透镜效应microturbulent velocity 微湍速度millimeter-wave astronomy 毫米波天文millisecond pulsar 毫秒脉冲星minimum mass 质量下限minimum variance 最小方差mixed-polarity magnetic field 极性混合磁场MMT, Multiple-Mirror Telescope 多镜面望远镜moderate-resolution spectrograph 中分辨摄谱仪moderate-resolution spectroscopy 中分辨分光modified isochrone method 改进等龄线法molecular outflow 外向分子流molecular shock 分子激波monolithic-mirror telescope 单镜面望远镜moom 行星环卫星moon-crossing asteroid 越月小行星morphological astronomy 形态天文morphology segregation 形态层化MSSSO, Mount Stromlo and Siding Spring Observatory斯特朗洛山和赛丁泉天文台multichannel astrometric photometer ( MAP )多通道天测光度计multi-object spectroscopy 多天体分光multiple-arc method 复弧法multiple redshift 多重红移multiple system 多重星系multi-wavelength astronomy 多波段天文multi-wavelength astrophysics 多波段天体物理naked-eye variable star 肉眼变星naked T Tauri star 显露金牛T 型星narrow-line radio galaxy ( NLRG ) 窄线射电星系Nasmyth spectrograph 内氏焦点摄谱仪natural reference frame 自然参考架natural refenence system 自然参考系natural seeing 自然视宁度near-contact binary 接近相接双星near-earth asteroid 近地小行星near-earth asteroid belt 近地小行星带near-earth comet 近地彗星NEO, near-earth object 近地天体neon nova 氖新星Nepturian ring 海王星环neutrino astrophysics 中微子天文NNTT, National New Technology Telescope国立新技术望远镜NOAO, National Optical Astronomical 国立光学天文台Observatories nocturnal 夜间定时仪nodal precession 交点进动nodal regression 交点退行non-destroy readout ( NDRO ) 无破坏读出nonlinear infall mode 非线性下落模型nonlinear stability 非线性稳定性nonnucleated dwarf elliptical 无核矮椭圆星系nonnucleated dwarf galaxy 无核矮星系nonpotentiality 非势场性nonredundant masking 非过剩遮幅成象nonthermal radio halo 非热射电晕normal tail 正常彗尾North Galactic Cap 北银冠NOT, Nordic Optical Telescope 北欧光学望远镜nova rate 新星频数、新星出现率NTT, New Technology Telescope 新技术望远镜nucleated dwarf elliptical 有核矮椭圆星系nucleated dwarf galaxy 有核矮星系number density profile 数密度轮廓numbered asteroid 编号小行星oblique pulsator 斜脉动星observational cosmology 观测宇宙学observational dispersion 观测弥散度observational material 观测资料observing season 观测季occultation band 掩带O-Ne-Mg white dwarf 氧氖镁白矮星one-parameter method 单参数法on-line data handling 联机数据处理on-line filtering 联机滤波open cluster of galaxies 疏散星系团Ophelia 天卫七optical aperture-synthesis imaging 光波综合孔径成象optical arm 光学臂optical disk 光学盘optical light 可见光optical luminosity function 光学光度函数optically visible object 光学可见天体optical picture 光学图optical spectroscopy 光波分光orbital circularization 轨道圆化orbital eccentricity 轨道偏心率orbital evolution 轨道演化orbital frequency 轨道频率orbital inclination 轨道倾角orbit plane 轨道面order region 有序区organon parallacticon 星位尺Orion association 猎户星协orrery 太阳系仪orthogonal transformation 正交变换oscillation phase 振动相位outer asteroid belt 外小行星带outer-belt asteroid 外带小行星outer halo cluster 外晕族星团outside-eclipse variation 食外变光overshoot 超射OVV quasar, optically violently OVV 类星体variable quasar、optically violent variable quasar oxygen sequence 氧序pan 摇镜头parry arc 彩晕弧partial-eclipse solution 偏食解particle astrophysics 粒子天体物理path of annularity 环食带path of totality 全食带PDS, photo-digitizing system、PDS、数字图象仪、photometric data system 测光数据仪penetrative convection 贯穿对流pentaprism test 五棱镜检验percolation 渗流periapse 近质心点periapse distance 近质心距periapsis distance 近拱距perigalactic distance 近银心距perigalacticon 近银心点perimartian 近火点period gap 周期空隙period-luminosity-colour relation 周光色关系PG 1159 star PG 1159 恒星photoflo 去渍剂photographic spectroscopy 照相分光。
AMOS词句中英文对照
AMOS词句中英⽂对照AMOS词句中英⽂对照王超整理Covariance 协⽅差（共变关系）Data Files 数据⽂件的连结设定File Manager ⽂件管理Interface Properties 界⾯属性Analysis Properties 分析属性Object Properties 对象属性Variables in Model 模型中的变量Variables in Dataset 数据⽂件中的变量Parameters 参数Diagram 绘图Draw Observed 描绘观察变量Draw Unobserved 描绘潜在变量Draw Path 描绘单向路径图Draw Covariance 描绘双向协⽅差图Figure Caption 图⽰标题（图形标题）Draw Indicator Variable 描绘指标变量Draw Unique Variable 描绘误差变量Zoom In 放⼤图⽰Zoom Out 缩⼩图⽰Loupe 放⼤镜检视Redraw diagram 重新绘制图形Identified 被识别unidentified ⽆法识别undo 撤销redo 恢复（重做）Copy to clipboafd 复制到剪切板Deselect all 解除选取全部对象Duplicate 复制对象Erase 删除对象Move Parameter 移动参数位置Reflect 映射指标变量Rotate 旋转指标变量Shape of Object 改变对象形状Space Horizontally 调整选取对象的⽔平距离Space Vertically调整选取对象的垂直距离Drag Properties 拖动对象属性Fit to Page 适合页⾯Touch up 模型图最适接触Model-Fit 模型适配度Calculate Estimates 计算估计值Stop Calculate Estimates停⽌计算估计值程序Manage Groups 管理群组/ 多群组设定Manage Models 管理模型/ 多重模型设定Modeling Lab 模型实验室Toggle Observed / Unobserved 改变观察变量/潜在变量Degree of Freedom ⾃由度的信息Specification Search 模型界定的搜寻Multiple-Group Analysis 多群组分析Bayesian estimation 适⽤于⼩样本的贝⽒估计法Data imputation 缺失值数据替代法List Font 字型Smart 对称性Outline 呈现路径图的线条Square 以⽅型⽐例绘图Golden 以黄⾦分割⽐例绘图Customize 定制功能列Seed Manager 种⼦管理Draw Covariances 描绘协⽅差双箭头图Growth Curve Model 增长曲线模型Name Parameters 增列参数名称Name Unobserved Variables 增列潜在变量名称Resize Observed Variables 重新设定观察变量⼤⼩Standardized RMR 增列标准化RMR值Plugins 增列Commands 命令Categories 分类Parameter Formats 参数格式Computation Summary 计算摘要Files in current directory ⽬前⽬录中的⽂件Standardized estimates 标准化估计Unstandardized estimates 未标准化估计View the input path diagram-Model specification显⽰输⼊的路径图View the output path diagram 显⽰输出结果的路径图Default model 预设模型Saturated model 饱和模型Independent model 独⽴模型1 variable is unnamed ⼀个变量没有名称Nonpositive definite matrices ⾮正定矩阵Portrait 肖像照⽚格式（纵向式的长⽅形：⾼⽐宽的长度长）Landscape 风景照⽚格式（横向式长⽅形：宽⽐⾼的长度长）Page Layout 页⾯配置Orientation ⽅向Apply 应⽤Latent variables 潜在变量Latent independent潜在⾃变量（因变量）Exogenous variables外因变量Latent dependent潜在依变量（果变量）Endogenous variables内因变量Draw a latent variable or add an indicator to a latent variable 描绘潜在变量或增画潜在变量的指标变量Rotate the indicators of a latent variable 旋转潜在变量的指标变量Error variable 误差变量Draw paths-single headed arrows 描绘单向箭头的路径Draw covariances-double headed arrows 描绘协⽅差（双向箭头）的路径Add a unique variable to an existing variable 增列误差变量到已有的变量中Residual variables 残差变量（误差变量）Minimization history 极⼩化过程的统计量Squared multiple correlations 多元相关平⽅/复相关系数平分Indirect, direct ＆ Total effects 间接效果、直接效果与总效果Sample moments样本协⽅差矩阵或称样本动差Implied moments 隐含协⽅差矩阵或称隐含动差Residual moments 残差矩阵或称残差动差Modification indices 修正指标Factor score weights 因素分数加权值Covariance estimates 协⽅差估计值Critical ratios for difference差异值的临界⽐值/ 差异值的Z检验Test for normality and outliers正态性与极端值的检验Observed information matrix 观察的信息矩阵Threshold for modification indices修正指标临界值的界定Means and intercepts 平均数与截距Page Setpage 设定打印格式Decimails⼩数点位数Column spacing 表格栏宽度Maximum number of table columns 表格字段的最⼤值Table Rules 表格范例Table Border 表格边框线Analysis Summary 分析摘要表Notes for Group 组别注解Fill color 形状背景的颜⾊Line width 边框线条的粗度Very Thin ⾮常细Very Thick ⾮常粗Fill style 填充样式Transparent 颜⾊透明Solid 完全填满Regular 正常字型Italic 斜体字型Bold 粗体字型Bold Italic粗斜体字型Set Default 设为默认值Set Default Object Properties 预设对象属性Pen width 对象框线Fill style 对象内样式Parameter orientation 参数呈现⽅向The path diagram 绘制的路径图中Normal template AMOS内定的⼀般样板格式中Visibility 可见性：显⽰设定项⽬在路径图上Use visibility setting 使⽤可见设置Show picture 显⽰图形对象Drag properties from object to object 将对象的属性在对象间拖动Height ⾼度X coordinate X坐标-⽔平位置Y coordinate Y坐标-垂直位置Parameter constraints 参数标签名称Preserve symmetries 保留对称性Zoom in on an area that you select 扩⼤选取的区域View a smaller area of the path diagram 将路径图的区域放⼤View a larger area of the path diagram 将路径图的区域缩⼩Show the entire page on the screen 将路径图整页显⽰在屏幕上Resize the path diagram to fit on a page 重新调整路径图的⼤⼩以符合编辑画⾯（路径图呈现于编辑窗⼝页⾯内）Examine the path diagram with the loupe 以放⼤镜检核路径图Multiple-Group Analysis 多群体的分析Specification Search 模型界定的搜寻Select one object at a time ⼀次选取单⼀对象Iteration 8 迭代次数为8Pairwise Parameter Comparisons 成对参数⽐较Varance-Covariance Matrix of Estimates 估计值间⽅差协⽅差矩阵Output输出结果标签钮Minimization history 最⼩化过程Standardized estimates 标准化的估计值Squared multiple estimates 多元相关的平⽅Indirect, direct ＆ total effects间接效果、直接效果与总效果Sample moments 观察样本协⽅差矩阵Implied moments 隐含协⽅差矩阵Residual moments 残差矩阵Modification indices 修正指标Tests for normality and outlies 检验正态性与异常值AMOS的五种选项估计法：Maximum likelihood 极⼤似然法，简称ML法Generalized least squares ⼀般化最⼩平⽅法，简称GLS法Unweighted least squares 未加权最⼩平⽅法，简称ULS法Scale-free least squares 尺度⾃由最⼩平⽅法，简称SFLS法Asymptotically distribution free 渐近分布⾃由法，简称ADF法“错误提⽰”部分：An error occurred while checking for missing data in the group, Group number 1.You have not supplied enough information to allow computing the sample variances and covariances. You must supply exactly one of the following: 没有提供⾜够的信息，因⽽⽆法计算样本的⽅差与协⽅差，使⽤者必须正确提供：a. The sample variance-covariance matrix. a. 样本⽅差-协⽅差矩阵b. The sample correlation matrix and the sample standard deviations b.样本相关矩阵与样本的标准差；c. Raw data. c.原始资料。
英译汉翻译历年真题及参考译文（4）
1997年英译汉试题及参考译⽂ Do animals have rights?This is how the question is usually put.It sounds like a useful，ground-clearing way to start.（71）Actually，it isnt，because it assumes that there is an agreed account of human rights，which is something the world does not have. On one view of rights，to be sure，it necessarily follows that animals have none.72）Some philosophers argue that rights exist only within a social contract，as part of an exchange of duties and entitlements.Therefore，animals cannot have rights.The idea of punishing a tiger that kills somebody is absurd，for exactly the same reason，so is the idea that tigers have ringhts.However，this is only one account，and by no means an uncontested one.It denies rights not only to animals but also to some people—for instance，to infants，the mentally incapable and future generations.In addition，it is unclear what force a contract can have for people who never consented to it:how do you reply to somebody who saysI dont like this contract? The point is this without agreement on the rights of people，arguing about the rights of animals is fruitless.（73）It leads the discussion to extremes at the outset:it invites you to think that animals should be treated either with the consideration humans extend to other humans，or with no consideration at all.This is a false choice.Better to start with another，more fundamental question:is the way we treat animals a moral issue at all? Many deny it.（74）Arguing from the view that humans are different from animals in every relevant respect，extremists of this kind think that animals lie outside the area of moral choice.Any regard for the suffering of animals is seen as a mistake—a sentimental displacement of feeling that should properly be directed to other humans. This view，which holds that torturing a monkey is morally equivalent to chopping wood，may seem bravelylogical.In fact it is simply shallow:the ethical equivalent of learning to crawl—is to weigh others interests against one s own.This in turn requires sympathy and imagination:without which there is no capacity for moral thought.To see an animal in pain is enough，for most，to engage sympathy.（75）When that happens，it is not a mistake:it is mankinds instinct for moral reasoning in action，an instinct that should be encouraged rather than laughed at. 1997年英译汉试题参考译⽂ 动物有权⼒吗?问题通常就是这样提出的。
霍金故事英语作文
霍⾦故事英语作⽂霍⾦是⼀个伟⼤的科学家，他的.故事⽆不感动着我们！下⾯店铺带来的是霍⾦故事英语作⽂，希望对你有所帮助！霍⾦故事英语作⽂1The way of life is progressive. It always goes along the incline of the infinite mental triangle. Nothing can stop him. What is life? Life is a very valuable thing, only once. Plants of different life can be regenerated, but human life can never be second times. There are many people who love life, such as: Zhang Haidi, rusty, etc.. Today, I want to introduce a man who loves life very much -- Hawking.Who is Hocking? A myth, a contemporary most outstanding theoretical physicist, a giant under the name of science. Maybe he is just a wheelchair and a brave man who challenges fate. Once, a wheelchair back to Berlin apartment, crossing the road was knocked down by a car, broken left arm, broken head, sewn 13 needles, about 48 hours later, he went back to the office to work. Though the handicap of the body is becoming more and more serious, Hocking tries to live like an ordinary person and accomplish anything he can do. He was even lively - it sounds a bit funny, and after he was completely unable to move, he still insisted on driving the wheelchair with the only movable finger to "hit" on the way to the office; when he met Prince Charles, he turned his wheelchair to show off, and the result rolled to Charles. The prince's toes. Of course, Hocking has also tasted the evil result of "free" action, the master of quantum gravity, many times in the weak gravity of the earth, jumping into a wheelchair. Fortunately, every time he was stubbornly "standing".In 1985, a tracheal surgeries were performed and the ability to speak was completely lost. Under such circumstances, he wrote the famous "brief history of time" very hard, and explored the origin of the universe.Though Hocking is a disabled person, he still loves life, tenacious struggle, and strives to go forward to the triangular slope. So what do we have to do with these non -disabled people? Do they love life more than they do?So called: life is like water. It is beautiful only when its rush and rush towards it.霍⾦故事英语作⽂2Stephen William Hocking was born in January 8th, and he graduated from the 31 School of University of Oxford and University of Cambridge. He received his Ph.D. in University of Cambridge. At the late stage of college learning, it began to suffer from amyotrophic lateral sclerosis (motor neuron disease) and hemiplegia. He overcame the difficulties with disabilities entered Gonville and Caius College of University of Cambridge in 1965 as a researcher. At this time, he studied the issue of the origin of the universe, and founded the famous theoretical year of the universe, "a point of infinite density", from the year of the outstanding scientific achievements of the College of Ren okunville and the College of Kai and Kai. He has been working at the Institute of astronomy, applied mathematics and theoretical physics at the University of Cambridge. Lecturer, Professor Lucas, Professor of mathematics. The year was elected as the youngest member of the Royal Society. It was the highest award of the world theory physics research award by the California Institute of Polytechnic Feil Child. Hocking's fame began with research on black holes. After Einstein merged another great theory of the twentieth Century, the other great theory, quantum theory, that the universe was limited, but could not find the margin, just as the earth's surface was limited but unable to find the margin; time was also beginning, and the Wolf prize in physics was awarded about 15 billion to 20 billion years ago.In 1985, Hocking lost his language ability and the only way to express his thoughts is a computer voice synthesizer. He uses a few movable fingers to manipulate a special mouse to select letters and words on a computer screen to make a sentence, and then play the sound through a computer, usually making a sentence for 5, 6 minutes. In order to make an hour's speech, he will be prepared to write a brief history of time in 10 days, until October 1995. The volume of the book has been over 25 million copies. It has been translated into dozens of languages, and the Chinese version has also been published.Author of "space - time large scale structure year and human co-author", "general relativity: Einstein Centennial review year", "superspace and supergravity year with man", "the beginning of the universe" and "a brief history of time"In 1990, he divorced his nurse, Elaine Mason, in 1990 when he divorced his wife, Jan Wilde, who had been married for 25 years.。
次时代纪元的英语
次时代纪元的英语In the era of tomorrow, where the sun rises on a horizon of endless possibilities, the English language has evolved into a tapestry of words so vibrant and colorful, it could make Shakespeare do a backflip and say, "Hold my quill!"Picture this: a world where "LOL" isn't just an acronym for laughter, but a universal sound that echoes through the cosmos, a symphony of joy that unites the galaxies. Where "selfie" is not just a word, but a form of time travel, capturing moments in the fabric of space-time, allowing us to revisit our past with a simple snap.In this futuristic epoch, the English language is notjust spoken, it's lived. It's a dance of words, a rhythmic beat that resonates with the pulse of the universe. It's a language that's as adaptable as a chameleon, changing hues to fit the mood of the moment, whether it's the electric buzz of a bustling metropolis or the serene whisper of a tranquil forest.Imagine a conversation that's more than just words; it's a performance, a spectacle where every sentence is a punchline, every dialogue a punch. A place where puns are the currency of the realm, and a well-timed joke can buy you a day's worth of smiles.And let's not forget the slang, oh the slang! It's thesecret sauce that gives our language its flavor. It's the secret handshake, the password to a club that's exclusive to those who are in the know. It's a language within a language, a code that only the initiated can decipher.In the next age, English is not just a means of communication; it's a tool for creativity, a canvas for our imaginations to run wild. It's a language that's as limitless as our dreams, as boundless as our aspirations.So, let's toast to the future, to the next chapter of the English language. May it continue to be the bridge that connects us, the glue that binds us, and the laughter that unites us. Cheers to the next era, where the only limit is our imagination, and the only rule is to have fun.。
应用本文方法求解几类常见的偏微分方程
分类号密级UDC 编号中国科学院研究生院硕士学位论文李群方法在求解几类偏微分方程中的应用何再明指导教师游亚戈研究员硕士中国科学院广州能源研究所申请学位级别硕士学科专业名称流体机械及工程论文提交日期2007.6论文答辩日期2007.6培养单位中国科学院广州能源研究所学位授予单位中国科学院研究生院答辩委员会主席Graduate School of Chinese Academy of SciencesThe application of Lie group to the solution of several classicalpartial differential equationsHe Zaiming (Fluid Machinery and Engineering)Directed by Prof. Y ou Y ageM.S.ThesisGuangzhou Institute of Energy Conversion, Chinese Academy of SciencesJune, 2007摘要传统能源供应日益紧张，开发可再生能源已成为当务之急。
海洋能是众多可再生能源的一种，波浪能利用是现今各国海洋能开发研究的重点，水动力学数值计算和非线性偏微分方程求解则是研究波浪能利用的工具。
在众多求解偏微分方程方法中，李群给我们提供了一种如何构造函数和自变量的变换方法，实现原微分方程的约化降维，求解此降维方程便获得原方程的解，它使人们在运用变换求解偏微分方程时变得有章可循。
本文的主要工作是，通过对求解偏微分方程的数值方法现状大致介绍和分析其自身局限性和缺陷，以李群理论的基础为始详细介绍了几类李群对称方法，之后以Mathematica这一科学计算软件为平台，通过对热传导方程的经典和非经典对称方法的推导，演示其推导方法和结果。
通过对其特征方程的常数赋予特值，获得其群不变量，再带回原方程实现降维约化，并求解该降维方程，从而获得热传导方程的解，然后对Burgers方程，Boussinesq方程，KdV方程不同方法求解来获得其更丰富的解。
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a rXiv:h ep-th/023149v116M ar22IFTUWr-03/2002/01CBPF-NF-001/02Generalized Space-time Supersymmetries,Division Algebras and Octonionic M-theory Jerzy Lukierski a ∗†and Francesco Toppan b ‡a Institute for Theoretical Physics,University of Wroc l aw,50-204Wroc l aw,pl.Maxa Borna 9,Poland b CBPF,CCP,Rua Dr.Xavier Sigaud 150,cep 22290-180Rio de Janeiro (RJ),Brazil Abstract We describe the set of generalized Poincar´e and conformal superalgebras in D =4,5and 7dimensions as two sequences of superalgebraic structures,taking values in the division algebras R ,C and H .The generalized conformal superalgebras are described for D =4by OSp (1;8|R ),for D =5by SU (4,4;1)and for D =7by U αU (8;1|H ).The relation with other schemes,in particular the framework of conformal spin (super)algebras and Jordan (super)algebras is dis-cussed.By extending the division-algebra-valued superalgebras to octonions we get in D =11an octonionic generalized Poincar´e superalgebra,which we call octonionic M-algebra ,describing the octonionic M-theory.It contains 32real supercharges but,due to the octonionic structure,only 52real bosonic generators remain independent in place of the 528bosonic charges of standard M -algebra.In octonionic M-theory there is a sort of equivalence between the octo-nionic M2(supermembrane)and the octonionic M5(super-5-brane)sectors.We also define the octonionic generalized conformal M-superalgebra,with 239bosonic generators.1IntroductionWe shall call generalized space-time supersymmetries the ones which go beyond the standard H L S scheme[1].In four dimensions using the framework of localfield theory and the arguments from S-matrix theory it was shown[1,2]that the bosonic sector B of Poincar´e or conformal superalgebra has the following tensor product structure1:B=B geom⊕B int,(1) where B geom describes space-time Poincar´e or conformal algebras and the remaining generators spanning B int are Lorentz scalars.It is easy to show that one can introduce the standard Poincar´e superalgebra,satisfying the relation(1),in any dimension(see e.g.[3]),but one arrives at a difficulty in trying to impose in any dimension standard conformal superalgebras. It appears[4,5,6]that one can introduce only at D=3,4and6an infinite series of standard conformal superalgebras,which can be denoted in a unified way as UαU(4;n|F)2[6,7](F=R for D=3,F=C for D=4and F=H for D=6).More explicitly3D=3:UαU(4;n|R)≡OSp(n;4|R),D=4:UαU(4;n|C)≡SU(2,2;n),D=6:UαU(4;n|H)≡O∗Sp(8;2n).It appears that if we wish to use the notion of conformal superalgebra in any dimension we should extend the standard Poincar´e superalgebra(see e.g.[10,11,12,13,14]).The best known case is in D=11,where the generalized Poincar´e superalgebra going beyond the H L S axioms is called the M-algebra and looks as follows(r,s=1,2,...,32;µ,ν=0,1,...,10):)rs Z[µ1,...,µ5],(2) {Q r,Q s}=Z rs=(CΓµ)rs Pµ+(CΓ[µν])rs Z[µν]+(CΓ[µ1 (5)where C=Γ0is the D=11real Majorana charge conjugation matrix.The generalized D=11 conformal superalgebra is obtained by adding a second copy of the superalgebra(2),with the extension of the conformal accelerations sector to the32×32symmetric matrices˜Z rs:)˜Z[µ1,...,µ5].(3) {S r,S s}=˜Z rs=(CΓµ)rs Kµ+(CΓ[µν])rs˜Z[µν]+(CΓ[µ1 (5)Both sets of generators Z rs,˜Z rs are Abelian,i.e.[Z rs,Z tk]=[˜Z rs,˜Z tk]=0.(4) It appears that if we introduce the crossed anticommutator,completing the superalgebra rela-tions{Q r,S s}=L rs,(5) O(5,2)×SU(2)(see e.g.[8,9]).we get from the Jacobi identity that the1024generators L rs form the GL(32;R)algebra[15]. Summarizing,the resulting superalgebra admits the followingfive-gradingI−2I−1I0I1I2˜Z rs S r L rs Q r Z rs.(6)The set of generators Z rs,˜Z rs,L rs describe the generalized D=11conformal algebra Sp(64) (conformal M-algebra)and all the generators from(6)form the superalgebra OSp(1|64)[16,17, 19,20],known as generalized D=11superconformal algebra(conformal M-superalgebra).The aim of this paper is to propose an analogous construction for the sequence of F-valued (F=R,C,H,O)generalized superalgebras,with the real superalgebras describing generalized supersymmetries in D=4.We shall describe these superalgebras in some detail in Sect.2 for D=4(F=R),5(F=C)and7(F=H).We obtain the generalized Poincar´e superalgebras with10real bosonic generators for D=4,16real bosonic generators for D=5and28real bosonic generators for D=7and the corresponding D=4,5and7generalized conformal superalgebras UαU(8;1|F).In Sect.3we shall consider the relation of our proposal to other ways of introducing generalized supersymmetries,in particular based on Lorentz spin and conformal spin algebras [21,22,23].It appears that our scheme for D=7can be identified with the one following from the minimal conformal spin algebra,but this is not the case for D=4,5.On the other hand our generalized superalgebras can be called minimal in another sense since the symmetrized product of supercharges(i.e.the anticommutators)is spanned by the fundamental representation of the respective Clifford algebras(R4×R4for D=4,C4×C4for D=5,H4×H4for D=7)4. The proposal is linked with the generalized conformal and superconformal algebra description in terms of F-valued Jordan(super)algebras[26,27,11].In Sect.4we shall conjecture that one can use the proposed superalgebras with the division algebra F given by the octonionic algebra O5.In particular we obtain in place of the“standard”M-algebra(2)an algebra which we call the octonionic M-algebra with52real bosonic generators described by a4×4octonionic Hermitian matrix.We provide two alternative descriptions of the octonionic M-algebra:thefirst one linear and bilinear in the octonionicΓ-matrices and the second with theirfive-linear products only.We shall also introduce an octonionic conformal M-superalgebra with232real bosonic generators.In Sect.5we present thefinal remarks.In particular we list some aspects of our framework which are postponed to further consideration.2The generalized D=4,5and7supersymmetries de-scribed by F-valued superalgebras(F=R,C,H)i)Generalized Poincar´e superalgebras.The standard N=1D=4Poincar´e superalgebra has the following complex Hermitian form(A,B=1,2):{Q A,4The fundamental representation of the Clifford algebra is its faithful representation with minimal dimension [24,25].5For the extension of UαU(n;m|F)algebra to octonions see also[28,29].{Q A,Q B}=0{Q˙}=0,(7)Bwhereσµ=(12,σi)describes the linear basis of Hermitian2×2matrices and Q A→Q A†=}=0,Q˙B{Q A,Q B}=Z AB,{Q˙}=B(σ[µν]AB Z AB−˜σ˙A˙B[µν]2i6F=C and H;see however Sect.4where we consider F=O.We add that an abstract algebra{A,B}=C of matrices having entries(a ij,b ij,c ij respectively)valued in a division algebra F(i.e.a ij= αa ijατα,withτ0the identity,and similarly for B,C),implies the following relations on the real components c ikγ= j,α,β{a ijα,b jkβ}Cαβγ,where Cαβγare the structure constants of F.where e r are the three quaternionic units.Then we shall consider O(4,1)and O(6,1)as D=3 and respectively D=5Euclidean conformal algebras and follow the rules(see e.g.[34])in order to introduce Dirac’sγmatrices for O(p+1,q+1)by doubling the dimension of the O(p,q)representations.We obtaini1)D=5.Γr(5)= 0σrσr0 ,Γ4(5)= 1200−12 ,Γ0(5)= 012−120 .(14) One can describe the complex Hermitian4×4matrices as linear combination of16Hermitian-symmetric matricesΓµ(5)C(5),Γµν(5)C(5)and iC(5).One sets{Q a,Q b†}=Z ab=(Γµ(5)C(5))ab Pµ+(Γ[µν](5)C(5))ab Z[µν]+iC(5)ab Z,(15) where C(5)is the O(4,1)complex charge conjugation matrix satisfying the relationsΓµ(5)†C(5)=−C(5)Γµ(5),C(5)†=−C(5).(16) In the representation withΓa(5)=Γa(5)†(a=1,2,3,4)(see e.g.(14))andΓ0(5)=−Γ0(5)†, we should put C(5)=Γ0(5).The maximal covariance algebra of the supercharges is given by the group GL(4,C),however distinguished role is played by its subgroup Uα(4;C)=U(2,2), because sixteen generators Z ab from(15)belong to the adjoint representation of U(2,2).i2)D=7.The Hermitian quaternionic representation of the O(6,1)Clifford algebra can be obtained from(14)as follows(p=1,...,5,µ=0,1, (6)Γp(7)= 0ΣpΣp0 ,Γ6(7)= 1200−12 ,Γ0(7)= 012−120 .(17) The O(6,1)quaternionic charge conjugation matrix C(7)satisfies relations analogous to(16) with quaternionic-Hermitian conjugation.In the representation withΓk(7)=Γk(7)†(k= 1,2,3,4,5,6)andΓ0(7)=−Γ0(7)†(see e.g.(17))we obtain again C(7)=Γ0(7).If we consider the symmetry properties of the products CΓ[µ1,...,µk](k=1,...,7)underquaternionic conjugation we obtain that(11)for F=H can be decomposed as follows(µ,ν= 0,1,...,6):{Q a,Q b†}=Z ab=(C(7)Γµ(7))ab Pµ+(C(7)Γ[µν](7))ab Z[µν].(18) The most general covariance group of quaternionic Poincar´e algebra(18)is GL(4,H),and its distinguished subgroup is Uα(4,H)≃SO∗(8)≃SO(6,2).The28bosonic real generators spanning Z ab in(18)are described by the adjoint representation of Uα(4,H).ii)Generalized conformal superalgebras.Following the procedure of obtaining OSp(1;64)from the M-algebra(2)one can add a second copy of the F-valued superalgebra(11){S a,S b}=˜Z ab(19)and impose the Jacobi identities which imply that the mixed anticommutator {Q a ,S b }=L ab describes the GL (4|F )Lie algebra generators.One obtains the following five-fold graded structureI −2I −1I 0I 1I 2˜Z ab S a L ab Q a Z ab .(20)If F =R the set of generators (20)describe the D =4generalizedconformalsuperalgebra OSp(1|8)with its bosonic sector describing the D =4generalized conformal algebra.The construction for D =5and D =7corresponds respectively to F =C and F =H .ii1)D =5In such a case the generalized conformal superalgebra is complex.The complex generators Z ab ,˜Zab in (20)describe complex Hermitian algebras (see (15)),and L ab span the GL (4|C )algebra.It can be checked that the complex bosonic algebra with three-gradingI −2I 0I 2˜X ab L ab X ab .(21)describes the U α(8,C )=U (4,4)algebra which is our D =5generalized conformal algebra.The five-grading (20)provides SU (4,4;1)as D =5generalized conformal superalgebra.ii2)D=7This case corresponds to inserting in (20)into the sectors I 2and I −2two copies of the D =7Poincar´e superalgebra given by (18).The sector I 0is then described by GL (4;H )≃SU ∗(8)algebra,and the quaternionic three-graded algebra (21)provides U α(8|H )≃O ∗(16)as the D =7generalized conformal algebra.The supersymmetric extension can be obtained by imposing the five-grading (20)and it leads to the D =7generalized conformal superalgebra U αU (8;1|H ).Summarizing we see that the D =4,D =5and D =7generalized conformal algebras and generalized conformal superalgebras are given respectively by U α(8|F )and U αU (8;1|F ).We obtain the following numbers of additional (in comparison with O (D,2))bosonic generators,which are present in generalized conformal algebras and conformal superalgebras:O (D,2)O (D,2)21(10)D =5(F =C )4384(28)standard space-timeD=44+6=105D=77+21=28(23)The field realizations on space-time with additional coordinates can be related with the repre-sentations of infinite-dimensional spin algebras with infinite spin or helicity spectra [20].3Relations with spinor algebras,representations of Clif-ford algebras and Jordan algebrasThe existence of standard conformal supersymmetries at D=3,4and6described by the set of superalgebras UαU(4;n,F)follows from the property that the spinorial coverings of the conformal algebra O(D,2)are described for D=3,4,6by Uα(4|F),i.e.Spin(3,2)=O(4,2)=O(6,2)=Uα(4|H)=O∗(8,C).(24) For D=5and D>6the spinorial covering ofO(n,m)8O(D,2),but for D=5and D≥7we obtain that dim Spin min(D,2)>dim O(D,2)=1are described by O(5,2)seven-vector,and afterO(5,2)dimensional reduction they will form an O(4,1)five-vector and two D=5scalars.These seven generators will extend the D=5conformal algebra O(5,2).In the supersymmetric case UαU(4;n|H)is used as D=5conformal spin superalgebra and will contain,besides the seven generators from O(6,2)7The fundamental spinor representation is determined by the minimal faithful Clifford algebra representation of O(n,m)with generators Iµν=1From the considerations in Sect.2follows that our D=5generalized conformal algebra SU(4,4)⊃Uα(4;H).Subsequently,for our D=5generalized conformal superalgebra we obtainSU(4,4;2)⊃UαU(4;1|H),(28) but UαU(4;1|H)is not contained in SU(4,4;1)in analogy with the relations between D=4 standard and generalized conformal superalgebras[35],where OSp(8;2)⊃SU(2,2;1),but one cannot embed SU(2,2;1)into OSp(8;1).It is easy to see that SU(4,4)in comparison with minimal D=5conformal spin algebra SU(4,4)⊃Uα(4;H)contains more additional bosonic generators.In fact the principle of constructing our generalized supersymmetries in D=4and D=5are analogous.In D=4 we relaxed the restrictions on superalgebra by replacing the complex structure by a real one, and in D=5the quaternionic structure is replaced by the complex one.ii)D=7.In D=7the situation is different.The fundamental D=7conformal spinors are given by H8and our generalized conformal superalgebra UαU(8;1,H)is identical with the minimal conformal spin superalgebraSpin min(7,2)=UαU(8;1|H)(29) with bosonic sector containing123bosonic generators(36O(7,2)generators+84additional tensorial generators+3generators describing U(1|H)=SU(2)R-symmetry).In order to compare the minimal conformal fundamental spin algebras[21,22,23]with our generalized conformal algebras let us write for D=4,5and7the fundamental Lorentz spin representations and minimal Clifford algebra modules permitting to represent faithfully O(D−1,1)Γ-matrices:min.Clifford mod.C(D−1,1)C2D=5C4H44Octonionic M-superalgebras and Octonionic M-theory One of the features of the proposed sequence of generalized supersymmetries is the possibility of extending the F-valued superalgebra structures to octonions.Octonions are described by eight real numbers(k=1, (7)X∈O:X=X0+X k t k,(31)where the seven octonionic units t k satisfy the nonassociative algebra1t k t l=−δkl+t k=−t k,and unit octonions describe the unit sphere S7through X10One can also say that S7describes the octonionic extension U(1|O)of U(1);S7is however not a Lie group, but rather the so-called soft Lie group[36].11Equivalently,to construct D=11octonionic gamma matrices we could start from the octonionic realization of Clifford algebra with(1,8)signature.It is worth mentioning that octonionic realizations of Clifford algebras only exist in specific signatures,such as(0,7),(9,0),(1,8),(10,1),(2,9)etc.They are related to the nonasso-ciative realizations of D=1N-extended supersymmetries(see[37]),which are beyond the classification of the representations of associative D=1N-extended supersymmetries[38],based on the Clifford algebras over R, C and H.From the multiplication table of the octonions follows that(k,l=1, (7)Γkl(1)=Γ[k(11)Γl](11)=f kl mΓm(11)Γ8(11)Γ9(11)Γ10(11)Γ11(11).(38) We see that out of the21bilinear products of thefirst seven matrices(36)only7are independent and they correspond to the generators of O(7)G2,which plays also the role of octonionic D=11Lorentzalgebra.It is easy to see that the maximal number of real generators on the r.h.s.of(33) is52=11+41,i.e.the relation(37)indeed saturates the octonionic-valued anticommutator {Q a,Q b†}(a,b=1,2,3,4).It should also be stressed that in the definition of n>2antisymmetric products of the octonionicΓ-matrices(36)one should provide the order of multiplication,because the Dirac algebra with the basis(36)is non-associative.To be explicit,the antisymmetrized product of n octonionic matrices A i(i=1,2,...,n)is given by[A1·A2·...·A n]≡12(.((A1A2)A3...)A n)+112The relation(38)is a particular case of the formula(42).13We note that in order to express the octonionic structure as constraints on the528real Abelian tensorial charges describing the generalized supersymmetry of standard M-theory we use the definition of octonionic-valued anticommutator from footnote6.a)The octonionic supermembrane is characterized by constrained number of two-tensorial charges-from55to41.The remaining14generators of G2describe inner automorphisms of the algebra(32)of octonionic units.b)If we keep the11degrees of freedom corresponding to the momentum sector,thefive-superbrane is also characterized by41independent degrees of freedom i.e.462degrees of freedom of standardfive-tensor charges in M-theory are restricted very much indeed.It is interesting tofind a geometric interpretation of such a huge reduction of degrees of freedom.c)The D=11Lorentz covariance algebra is described also by41generators of O(10,1)G2,replacing standard D=11conformal algebra.b)The128real generators of Gl(4|O)describe the4×4octonionic matrix in the place of the general real Gl(32|R)covariance group of standard M-theory with1024real generators.iii)The octonionic conformal M-superalgebra.The superextension of UαU(8;1|F)to F=O describes the octonionic conformal M-super-algebra,with bosonic sector described by Uα(8|O)×U(1|O)(232+7=239real generators), where the internal sector U(1|O)≃S7describes a parallelizable manifold which only can be described by an extension of the notion of standard Lie algebra-the so called soft Lie algebras [36].Some indications suggest that the structure of the octonionic supergroups UαU(4;1|O) and UαU(8;1|O)in the real basis is that of a(graded)Malcev(super)algebra[39,40].We leave this investigation for future work.5Concluding remarksOur proposal is the extension of Kugo and Townsend[5]relation between division algebras and standard sequence D=3,4,6,10of supersymmetries within the H L S scheme to the case of generalized supersymmetries in the dimensions4,5,7and11.The idea that d-dimensional Minkowski space-time should be extended by additional dimensions,describing tensorial central charges coordinates,has been proposed already some time ago(see e.g.[41,42,43,44]).Our framework provides a concrete way of extending standard space-time framework to dimensions 5and7and ultimately to D=11.In this paper we did not develop various aspects of the proposed scheme.Let us only present a list of them,as problems for possible further considerations.i)One can ask if the choice of our sequence and its space-time supersymmetry interpretation is unique.Indeed,because UαU(8;1|C)≡SU(4,4;1)include D=6conformal symmetries O(6,2),one could also assign our sequences of superalgebras to D=4,6and7.The argument for using the sequence D=4,5,7comes from the link with minimal Clifford algebra realizations (see(33)).The other choice of division superalgebra sequence can be obtained if we replace the quaternionic structure of D=6Poincar´e superalgebra by the real one14.In such a case one obtains the sequence UαU(16;1|F)as describing generalized conformal superalgebras in D=6 (OSp(1;16|R)),D=7(SU(8,8:1))and D=9(UαU(16;1|H)).ii)For simplicity we do not consider here more explicitly the extended generalized symme-tries,but such a generalization is obvious.In particular the extended generalized conformal supersymmetry with N copies of of F-valued supercharges Q a i,S a i is given by the superalge-bra UαU(8;N|F),with the internal sector(R–symmetries)U(F)(O(N)for F=R,U(N)for F=C and U(N;H)≡USp(2N)for F=H).iii)Our considerations are on purely algebraic level.One should also consider the represen-tation theory of the generalized(super)symmetry algebras,e.g.express the generators in terms of oscillators15and consider the complete set of Casimir’s.iv)We did not mention here generalized de-Sitter supersymmetries and mentioned only for F=O anti-de-Sitter symmetries and supersymmetries.The generalized anti-de-Sitter super-algebras in dimension D should be identified with the generalized conformal superalgebras in dimension D−1,i.e.our set of superalgebras UαU(8;1|F)describes the generalized anti-de-Sitter algebras in D=5,6,8and possibly D=12(the last case for F=O).The discussion of generalized D-dimensional de-Sitter superalgebras,equivalent to generalized Lorentz superal-gebras in dimension D+1is similar in principle,however with differences in technical details (see also[14]).v)We did not discuss here an important issue of supersymmetric dynamics,covariant un-der generalized Poincar´e and conformal supersymmetries.We would like only to mention that the preliminary results in such a direction has been already presented[18,45,19]for massless D0-superbranes(supersymmetric particles)mainly in D=4with OSp(1;8)as generalized conformal algebra.AcknowledgmentsThe authors would like to thank Dmitri Sorokin for valuable comments.One of the authors(J.L.)would like to thank Jos´e Helayel-Neto for his hospitality at CBPF in Rio de Janeiro,where the main part of this paper was prepared.F.T.is grateful for the hospitality at the Institute of Theoretical Physics of the University of Wroc l aw,where this work has been completed. 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