Constraints on Bars in the Local Universe from 5000 SDSS Galaxies
Obtaining a Converged Solution with Abaqus_L6
Convergence Problems:Element BehaviorLecture 6L6.2Overview•Hourglassing in Reduced-Integration Elements•Checkerboarding•Ill-ConditioningObtaining a Converged Solution with Abaqus163Hourglassing in Reduced-IntegrationElementsL6.4 Hourglassing in Reduced-Integration Elements•Full integration vs. reduced-integration elements•In elastic finite element analysis the strain energy density must beintegrated over the element volume to obtain the element stiffnessmatrix.•Full integration refers to the minimum Gauss integration orderrequired for exact integration of the strain energy (if the element isnot distorted).•Reduced integration refers to a Gauss integration rule of one orderless than full integration.•The reduced integration method can be used only in quadrilateraland hexahedral elements.Obtaining a Converged Solution with Abaqus164•What is hourglassing?•The use of the reduced-integrationscheme has a drawback: it can resultin mesh instability, commonly referredto as “hourglassing.”•Consider a rectangular plate simplysupported along two edges.•The hourglass mode does not causeany strain and, hence, does notcontribute to the energy integral.•It behaves in a manner that issimilar to that of a rigid bodymode.Obtaining a Converged Solution with AbaqusL6.6 Hourglassing in Reduced-Integration Elements•The hourglass mode insecond-order “serendipity”(8-node) elements (CPS8R,CPE8R) is nonpropagating.•Neighboring elements cannotshare the mode, so the modecannot occur in a mesh withmore than two elements.•There is no real danger ofhourglassing in theseelements.Obtaining a Converged Solution with Abaqus165•Hourglass modes of first-orderreduced-integration quadrilateral andhexahedral elements can propagate;therefore, hourglassing can be aserious problem in those elements.•To suppress hourglassing, anartificial “hourglass control”stiffness must be added.•Abaqus uses hourglass control inall first-order reduced-integrationelements.Obtaining a Converged Solution with AbaqusL6.8 Hourglassing in Reduced-Integration Elements•General comments•Hourglassing is mainly an issue with first-order reduced-integrationquad/hex elements.•Regular triangular and tetrahedral elements in Abaqus always usefull integration and, hence, are not susceptible to hourglassing.•Hourglassing can occur in geometrically linear and geometricallynonlinear problems, including finite-strain problems.•In geometrically linear problems hourglassing usually does not affect thequality of the calculated stresses.Obtaining a Converged Solution with Abaqus166Obtaining a Converged Solution with Abaqus •In geometrically nonlinear analysis the hourglass modes tend to interact with the strains at the integration points, leading to inaccuracy and/or instability.•Hourglassing is particularly troublesome for problems involving finite-strain elasticity (hyperelasticity) or very large (incompressible) plastic deformations.•Fully integrated elements are strongly recommended, whenever feasible, for finite-strain elasticity analysis.•When hourglassing is creating convergence problems, the simulation will often have many diverging solution cutbacks.L6.10Obtaining a Converged Solution with AbaqusHourglassing in Reduced-Integration Elements•When is hourglassing a problem?•Hourglassing is almost never a problemwith the enhanced hourglass controlavailable in Abaqus.•More robust than other schemes•No user-set parameters•Based on enhanced strain methodsRubber disk rollingagainst rigid drumCombined hourglass control scheme Enhancedhourglass controlscheme l o a dd i sk r o ll dr uml o a d d i s k roll dru mComparison of energy historiesALLIE ALLIEALLAE ALLAE167•Currently, enhanced hourglass control is not the default scheme formost elements.•The following table summarizes the hourglass control methods currentlyavailable in Abaqus, including the default schemes for most elements:Abaqus/Standard Abaqus/ExplicitStiffness (default)Relax stiffness (default)Enhanced strain Enhanced strainStiffnessViscousCombined (stiffness+viscous)•But…enhanced strain hourglass control is the default for:•All modified tri and tet elements•All elements modeled with finite-strain elastic materials(hyperelastic, hyperfoam, and hysteresis)Obtaining a Converged Solution with AbaqusL6.12 Hourglassing in Reduced-Integration Elements•To activate enhanced hourglasscontrol, use the option*SOLID SECTION, CONTROLS=name,ELSET=elset*SECTION CONTROLS, NAME=name,HOURGLASS=ENHANCEDNo user parameters•Abaqus/CAE usage:Mesh module: Mesh→Element TypeObtaining a Converged Solution with Abaqus168Obtaining a Converged Solution with Abaqus •Detecting and controllinghourglassing•Hourglassing can usually be seenin deformed shape plots.•Example: Coarse and mediummeshes of a simply supportedbeam with a center point load.•Excessive use of hourglass controlenergy can be verified by looking atthe energy histories. •Verify that the artificial energy used to control hourglassing is small (1) relative to theinternal energy.Same load and displacement magnification (1000×)L6.14Obtaining a Converged Solution with AbaqusHourglassing in Reduced-Integration Elements•Use the X –Y plotting capability in Abaqus/Viewer to compare the energies graphically.Internal energy Artificial energy Artificial energyInternal energyTwo elements through the thickness: Ratio of artificial to internal energy is 2.Four elements through the thickness: Ratio of artificial to internal energy is 0.1.169Obtaining a Converged Solution with Abaqus •Example: Engine mount•Consider two forms ofhourglass control:•Stiffness-based•Enhanced strain Outer rim moves up under load controlDISPLACEMENT MAGNIFICATION FACTOR = 1.00 ORIGIRESTART FILE = a STEP 1 INCREMENT 20TIME COMPLETED IN THIS STEP .267 TOTAL ACCUMULATED TIMEABAQUS VERSION: 5.6-4 DATE: 15-MAY-97 TIME: 13:16:10123steelrubber L6.16Obtaining a Converged Solution with Abaqus GNIFICATION FACTOR = 1.00 ORIGINAL MESH DISPLACED MESH a STEP 1 INCREMENT 20IN THIS STEP .267 TOTAL ACCUMULATED TIME .267Nonconvergence at 27of load Hourglassing in Reduced-Integration Elements•Results with stiffness hourglass controlDISPLACEMENT MAGNIFICATION FACTOR = 1.00RESTART FILE = a STEP 1 INCREMENT 20TIME COMPLETED IN THIS STEP .267 TOTAL ACCUMULATED TIME .267ABAQUS VERSION: 5.6-4 DATE: 15-MAY-97 TIME: 13:16:10123Severe hourglassing occurs 170Obtaining a Converged Solution with Abaqus •Results with enhanced hourglass controlGNIFICATION FACTOR = 1.00 ORIGINAL MESH DISPLACED MESHa2 STEP 1 INCREMENT 13IN THIS STEP 1.00 TOTAL ACCUMULATED TIME 1.00 5.6-4 DATE: 15-MAY-97 TIME: 13:26:19Deformation at 100of load; rubber uses default enhanced hourglass controlNo hourglassing DISPLACEMENT MAGNIFICATION FACTOR = 1.00RESTART FILE = a2 STEP 1 INCREMENT 13TIME COMPLETED IN THIS STEP 1.00 TOTAL ACCUMULATED TIME 1.00ABAQUS VERSION: 5.6-4 DATE: 15-MAY-97 TIME: 13:26:19123L6.18Obtaining a Converged Solution with AbaqusHourglassing in Reduced-Integration Elements •Elastic bending problems and coarse mesh accuracy•For elastic bending problems, improved coarse mesh accuracy may be obtained using the enhanced hourglass control method.•The enhanced hourglass control formulation is tuned to giveaccurate results for regularly shaped elements undergoing elastic bending.•Where these conditions apply, a coarse mesh may give acceptable results despite the artificial energy being greater than a few percent of the internal energy.•An independent check of the results should be made to determine if they are acceptable.171•Plastic bending problems•When plasticity is present, the stiffness-based hourglass control causeselements to be less stiff than in the enhanced control case.•This may give better results with plastic bending; enhancedhourglass control may cause delayed yielding or excessivespringback.•In using enhanced hourglass control in this case, the usual rule-of-thumb regarding the acceptable level of artificial energy should befollowed.•Recall that C3D10M elements use enhanced hourglass control bydefault.•Use alternative hourglass control for problems involvingyielding.Obtaining a Converged Solution with AbaqusCheckerboarding172•Whereas hourglassing is a behavior where large, oscillatingdisplacements occur without significant stresses, checkerboarding is abehavior where large, oscillating stresses occur without significantdisplacements.•Checkerboarding typically occurs for hydrostatic stresses in (almost)incompressible materials that are highly confined.•It can occur in first-and second-order elements but is most notablein first-order elements.•It is more likely to occur in regular meshes than in irregular meshes.Obtaining a Converged Solution with AbaqusL6.22 Checkerboarding•Checkerboarding is related to—but is not the same as—volumetriclocking.•Volumetric locking occurs when incompressible material behaviorputs more constraints on the deformation field then there aredisplacement degrees of freedom.•For example, in a refined, three-dimensional mesh of 8-nodehexahedra, there is—on average—1 node with 3 degrees offreedom per element.•The volume at each integration point must remain fixed. Sincefull integration uses 8 points per element, we have as many as8constraints per element but only 3 degrees of freedom.•Consequently, the mesh is overconstrained—it locks.•Volumetric locking can be avoided by using the proper elementtype; for a more detailed discussion of this topic see the “ElementSelection in Abaqus” lecture notes.Obtaining a Converged Solution with Abaqus173•Checkerboarding does not always manifest itself clearly.•The displacement field may initially be unaffected, and stresscontour plots may not show the checkerboarding because ofsmoothing of element stresses during postprocessing.•Discontinuous or “quilt” plots will show the checkerboardpattern, however.•In linear analyses checkerboarding rarely causes convergencedifficulties.•However, in nonlinear analyses the high hydrostatic stressoscillations can eventually interact with the displacements andcause sudden, usually catastrophic, convergence problems.•Checkerboarding can be eliminated by introducing some local meshirregularities.Obtaining a Converged Solution with AbaqusL6.24 Checkerboarding•Example: Rubber bushing•Consider a cylindrical rubber bushing made of an (almost)incompressible rubber.•The bushing is modeled with first-order, generalized plane strainelements.•Both the inner and outer radius of the bushing are fully constrained.•This constraint severely limits the deformations that can occurin the model.• A compressive axial load is applied to the bushing through theelement reference node.Obtaining a Converged Solution with Abaqus174•In this model the element indicatedin the figure is given a bulk modulusthat is one order-of-magnitudesmaller than that assigned to therest of the elements in the mesh.•This should lead to a smallerhydrostatic pressure in thiselement.smaller KObtaining a Converged Solution with AbaqusL6.26 Checkerboarding• A “quilt” contour plot (withoutaveraging between elements)clearly shows a “checkerboard”pattern with a significant pressurevariation.Quilt (nonaveraged) contour plotof hydrostatic pressureObtaining a Converged Solution with Abaqus175Ill-ConditioningL6.28 Ill-Conditioning•When elements or materials show large stiffness differences,conditioning problems may occur.•In linear analyses problems typically occur only when the stiffnessdifferences are extreme (factors of 106or more).•In such cases the solution of the linear equation systembecomes inaccurate.•In nonlinear analyses problems occur at a much earlier stage.•The stiffness differences may cause poor convergence or evendivergence if the increment size is not very small.Obtaining a Converged Solution with Abaqus176•Long, slender or rigid structures•Large differences in stiffness occur in long, slender structures (such asvery long pipes or cables) or very stiff structures (such as a link in avehicle’s suspension system).•If such structures undergo large motions in geometrically nonlinearanalyses, convergence can be very difficult to obtain.•Slight changes in nodal positions can cause very large (axial) forcesthat, in turn, cause incorrect stiffness contributions.•This makes it very difficult or impossible for the usual finite elementdisplacement method to converge.•Convergence problems in these simulations usually manifest themselvesin very slow or irregular convergence rates or in diverging solutions.Obtaining a Converged Solution with AbaqusL6.30Ill-Conditioning•Use hybrid beam elements (types B21H, B31H, B31OSH) or hybrid trusselements to model such problems.•In these hybrid elements the axial and, in the case of hybrid beams, thetransverse shear forces in the elements are included as primaryvariables in the element formulation•Because the forces are primary variables, they remain reasonablyaccurate during iteration, and the elements usually converge faster.•Even though the additional primary variables make these elementsmore expensive per iteration, they are usually much more efficientbecause the improved convergence rate reduces the number ofiterations.Obtaining a Converged Solution with Abaqus177Obtaining a Converged Solution with Abaqus •Example: Near bottom pipeline pull-inand tow•Simulating a seabed pipelineinstallation.•Drag chains used to offsetbuoyancy effects.•Model:•Pipeline modeled using beamelements.•Pipeline is very slender.•One end of the pipeline iswinched into an anchor point.•The other end is built in for thepull-in and free for the tow.Pipeline dimensions:Length = 1000 ft Outer diameter = 0.75 ft Wall thickness = 0.025 ftL6.32Obtaining a Converged Solution with AbaqusIll-Conditioning•Pull-in analysis•The job with hybrid elements converges significantly faster than the job without hybrid elements.Element typeNumber of increments Number of cutbacks Number of iterations B33H26299B33281151178Obtaining a Converged Solution with Abaqus •Tow analysis•The pipeline has no restraint (and is, therefore, singular) until the drag chain extends sufficiently to stabilize the pipeline.INCREMENT 1 STARTS. ATTEMPT NUMBER 1, TIME INCREMENT 1.000E-02***WARNING : SOLVER PROBLEM. NUMERICAL SINGULARITY WHEN PROCESSING NODE 3D.O.F. 6 RATIO = 7.26788E+15***WARNING : THE SYSTEM MATRIX HAS 1 NEGATIVE EIGENVALUES .•To overcome numerical difficulties in the early stages of the analysis, a small initial stress is applied to the pipeline:*INITIAL CONDITIONS,TYPE=STRESS BEAMS,1.E-8Element type Number of increments Number of cutbacks Number ofiterationsB33H 260182B33372323L6.34Obtaining a Converged Solution with AbaqusIll-Conditioning•Approximately incompressible material behavior•If the bulk modulus, K , is much larger than the shear modulus, G , large stiffnesses occur inside an element.•Slight changes in nodal positions can cause very large volumetric strains and, as a result, large hydrostatic stresses.•The large hydrostatic stresses cause incorrect stiffnesscontributions, which seriously hamper convergence.•This effect is particularly seen with hyperelastic materials.179•Use hybrid solid elements (types CPE4H, C3D20H, CAX4H, etc.) insuch cases.•In these elements the hydrostatic pressure (or in some cases the volumechange) is included as a primary variable in the element formulation.•Consequently the hydrostatic stresses (and, thus, the effectivestiffness) remain reasonably accurate during the iteration process.•Although the cost per iteration increases due to the additionaldegrees of freedom, the overall analysis cost typically is reducedbecause a smaller number of iterations will be needed.•An exception is the modified 10-node tetrahedral elements(C3D10MH).•For these elements the cost per iteration increases significantly.Obtaining a Converged Solution with Abaqus180。
COBE Constraints on Baryon Isocurvature Models
1. Introduction
The original baryon isocurvature scenario for structure formation (Peebles 1987a,b) presents a simple and attractive alternative to the standard cold dark matter (CDM) cosmogony. It simultaneously satis es observations which require a low density universe 0 ' 0:2 0:3 (e.g. Dekel et al. 1993), forms structure without the aid of hypothetical dark matter, and can alter light element nucleosynthesis su ciently to make an 0 = baryonic universe acceptable (Gnedin & Ostriker 1992). Moreover recent measurements of a large Hubble constant H0 = 100h km s 1 Mpc 1 , h = 0:80 0:17 (Freedman et al. 1994) would be easier to accommodate in such a low density universe. Unfortunately, when normalized to the COBE DMR detection (Smoot et al. 1992), the open universe manifestations of this model appear to be inconsistent with several observations of cosmic microwave background anisotropies at degree (Chiba, Sugiyama, & Suto 1994; Hu & Sugiyama 1994, hereafter HS94), and arcminute scales (Efstathiou & Bond 1987; Hu, Scott, & Silk 1994). The model generically su ers from excess small scale power. However, given the present uncertain status of CMB anisotropy detections at degree to arcminute scales (see e.g. Wilkinson 1994), it is perhaps premature to rule out models on these grounds. The excess small scale power required in this scenario also appears as a steep slope in the large angle anisotropy spectrum (Sugiyama & Silk 1994). This prediction con icts with the at spectrum measured by the COBE DMR experiment (Gorski et al. 1994; Bunn, Scott, & White 1995). In this Letter, we quantify this constraint on the standard isocurvature baryon model by employing the techniques of Bunn & Sugiyama (1995) to analyze the two year COBE DMR maps. Unlike previous treatments (Chiba, Sugiyama, & Suto; HS94), we also use the full information in the COBE sky maps to determine the normalization as opposed to merely the rms uctuation at 10 degrees. This causes a 10% boost in the amplitude of uctuations in open models. However we further extend prior treatments by considering at low 0 , cosmological constant models whose predictions are somewhat more in accord with observations. The boost in amplitude can be up to 30% in these models. The corresponding change in the matter uctuation amplitude 8 is relevant for simulations of large scale structure formation. Finally, employing spectral distortion constraints from the COBE FIRAS experiment (Mather et al. 1994), we nearly close the parameter space available to these baryon isocurvature models. For the small class of models remaining, we present the predictions for degree scale anisotropies and large scale structure.
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（1）He is best known for coining the term fractal to describe phenomena (such as coastlines, snowflakes, mountains and trees) whose patterns repeat themselves at smaller and smaller scales.(2) To explain these puzzling findings, some scientists have revived an old idea of Einstein’s that had been discarded as false: that the vacuum of space has energy in it that acts repulsively and accelerates the expansion of the universe.(3) New language that seemed to assert Facebook’s “irrevocable”right to retain and use a member’s personal information, even after the member had closed his or her Facebook account, deserved a little more editing.(4) The tenor of our time appears to regard history as having ended, with pronouncements from many techno-pundits claiming that the Internet is revolutionary and changes everything.(5) As a by-product of an industry that exists all over the world-the stalks that remain after grain has been harvested-straw also helpfully soaks up carbon from the atmosphere and locks it in, so long as it is not allowed to decompose.（1）The problem in the paradox, he reasoned, is that we are confusing a description of sets of numbers with a description of sets of sets of numbers.（2）Now the cosmological constant is one of the leading theories of why the universe is blowing up like a balloon at ever-increasing speed.（3）That’s not because Facebook is hell-bent on stripping away privacy protections, but because the popularity of Facebook and other social networking sites has promoted the sharing of all things personal, dissolving the line that separates the private from the public.（4）Although it’s unclear if the capacity for remote operation will justify the QB’s US$15,000 price tag, Anybots believes its technology will appeal to a new generation of workers who expect to be in contact at all times and in all places.（5）As a by-product of an industry that exists all over the world-the stalks that remain after grain has been harvested-straw also helpfully soaks up carbon from the atmosphere and locks it in, so long as it is not allowed to decompose.（1）But Russell ( and independently, Ernst Zermelo ) noticed that x = {a: a is not in a} leads to a contradiction in the same way as the description of the collection of barbers.（2）An analogy is the siren of an ambulance that changes pitch as it moves toward you, then passes you and heads the other way — its waves are first compressed, then stretched.（3）It’s the older members — those who could join only after it opened membership in 2006 to workplace networks, then to anyone — who are adjusting to a new value system that prizes self-expression over reticence. （4）Although it’s unclear if the capacity for remote operation will justify the QB’s US$15,000 price tag, Anybots believes its technology will appeal to a new generation of workers who expect to be in contact at all times and in all places.（5）No matter how stylishly modern and environmentally friendly straw-bale house may look, however, you still want to know that it won’t get sopping wet in a thunderstorm or go up in a whoosh of flames if you knock over a candle.（1）For example, we can describe the collection of numbers 4, 5 and 6 by saying that x is the collection of integers, represented by n, which are greater than 3 and less than 7.（2）Dark energy is the name given to an unexplained force that is drawing galaxies away from each other, against the pull of gravity, at an accelerated pace.（3）Among members, a Law of Amiable Inclusiveness seems to be revealing itself: over time, many are deciding that the easiest path is to routinely accept “friend requests”, completing a sequence begun when one member seeks to designate another as a Facebook friend.（4）The area between the QB’s head and base consists of a length of telescoping plastic that can be adjusted to let the QB stand as tall as 175 centimeters or as short as 81 centimeters.（5）After waiting another 45 minutes and finding that the panel still hadn’t failed, the team gave up and stopped the experiment, secure in the knowledge that the material had performed beyond the requirements of building regulations.Part III. Translate the following sentences into Chinese (20%) 1. This system served as vehicle for the first formalizations of the foundations of mathematics; it is still used in some philosophical investigations and in branches of computer science.2. That’s not because Facebook is hell-bent on stripping awayprivacy protections, but because the popularity of Facebook and other social networking sites has promoted the sharing of all things personal, dissolving the line that separates theprivate from the public.3. No matter how stylishly modern and environmentally friendlystraw-bale house may look, however, you still want to know that it won't get sopping wet in a thunderstorm or go up in a whoosh of flames if you knock over a candle.4. The premise on which the technology is based is that geneticinformation[1], encoded by DNA and arranged in the form of genes, is a resource that can be manipulated in various ways to achieve certain goals in both pure and applied science and medicine.5. Many people are working towards trying to make cities more sustainable. A sustainable city offers a good quality of life tocurrent residents but doesn’t reduce the opportunities for future residents to enjoy.Part III. Translate the following sentences into Chinese (20%)1. This system served as vehicle for the first formalizations of the foundations of mathematics; it is still used in some philosophical investigations and in branches of computer science.2. Dark energy is the name given to an unexplained force that is drawing galaxies[1] away from each other, against the pull of gravity, at an accelerated pace.3. When the distinction blurs between one’s few close friends and the many who are not, it seems pointless to distinguish between private and public.4. No matter how stylishly modern and environmentally friendly straw-bale house may look, however, you still want to know that it won't get sopping wet in a thunderstorm or go up in a whoosh of flames if you knock over a candle.5. Engineers in the future will work at the extremes of very large and very small systems that require greater knowledge andcoordination of multidisciplinary and multi-scale engineering across greater distances and timeframesPart III. Translate the following sentences into Chinese（20%）1. To explain the puzzling findings, some scientists have received an old idea of Einstein’s that had been discarded as false: that the vacuum of space has energy in it that acts repulsively and accelerates the expansion of the universe.2. Telecommuting workers and traveling executives alike could use QB( so named because it is the next in line after the company’s prototype QA bot ) as a virtual extension of themselves, allowing them to attend meetings, tour facilities or perform walk-throughs of real estate properties, all while controlling the robot from a computer keyboard.3. Research had developed to the point where progress was being hampered by technical constraints, as the elegant experiments that had helped to decipher the genetic code could not be extended to investigate the gene in more detail.4. The ideas that the state of the universe at one time determines the state at all other times, has been a central tenet of science, ever since Laplace’s time.5. Nuclear power’s prominence as a major ene rgy source will continue over the next several decades, according to new projections made by the International Atomic Energy Agency (IAEA), which has just published a new report, Energy, Electricity and Nuclear Power for the period up to 2030.Part III. Translate the following sentences into Chinese（20%）1. For a smallish two-bed roomed house with one largeopen-plan kitchen/dinner, that doesn’t particularly cheapgiven that straw is supposed to be inexpensive, and you’dstill have to buy the plot and dig the foundations.2. Combined with advances in CAD systems, it will be possible formechanical engineers to collaborate in immersive interactive environments where they can design collaboratively, testhypotheses, run models and simulations and observe theircreations in three dimensions much as an engineer canobserve a car being built with their colleagues on the shopfloor.3. Although it is unclear if the capacity for remote operation willjustify the QB’s US$15,000 price tag, Anybots believes itstechnology will appeal to a new generation of workers whoexpect to be in contact at all times and in all places,4. That’s not because Facebook is hell-bent on stripping awayprivacy protections, but because the popularity of Facebook and other social networking sites has promoted the sharing of all things personal, dissolving the line that separates theprivate from the public.5. Because gravity draws mass together, most experts expected to find that gravity had slowed down the universe’s rate of ballooning, or perhaps that the rate was staying about the same.Part III. Translate the following sentences into Chinese（20%）1. Dark energy is the name given to an unexplained force that is drawing galaxies away from each other, against the pull of gravity, at an accelerated pace. Dark energy is a bit likeanti-gravity. Where gravity pulls things together at the more local level, dark energy tugs them apart on the grander scale. 2. The growth of membership and of individual networks seems impervious to gaffes at the company during its brief, five-year history. One of those instances was in February, when it fiddled with its terms of service. New language that seemed to assert Facebook’s “irrevocable” right to retain and use a member’s personal information, even after the member had closed his or her Facebook account, deserved a little more editing.3. Along with the wheels, a self-balancing system and a motor with a top speed of five kilometers per hour make the robot mobile. The two-wheel—as opposed to a tricycle or quad—design makes it more maneuverable in tight spaces and helps keep its weight down to about 16 kilograms. The area between the QB's head and base consists of a length of telescoping plastic that can be adjusted to let the QB stand as tall as 175 centimeters or as short as 81 centimeters.4. Social and demographic changes are leading to a greater demand for housing. People are living longer, and choosing to marry later, and in recent years there has been a rise in thenumber of single-parent families. Added to this, the UK is experiencing immigration from other countries, e.g. from Poland[1]which has recently joined the EU. The result is an ever-larger number of smaller households, all requiring accommodation.Part III. Translate the following sentences into Chinese（20%）1. Dark energy is the name given to an unexplained force that is drawing galaxies away from each other, against the pull of gravity, at an accelerated pace. Dark energy is a bit likeanti-gravity. Where gravity pulls things together at the more local level, dark energy tugs them apart on the grander scale. 2. The growth of membership and of individual networks seems impervious to gaffes at the company during its brief, five-year history. One of those instances was in February, when it fiddled with its terms of service. New language that seemed to assert Facebook’s “irrevocable” right to retain and use a member’s personal information, even after the member had closed his or her Facebook account, deserved a little more editing.3. Along with the wheels, a self-balancing system and a motorwith a top speed of five kilometers per hour make the robotmobile. The two-wheel—as opposed to a tricycle orquad—design makes it more maneuverable in tight spaces and helps keep its weight down to about 16 kilograms. The areabetween the QB's head and base consists of a length oftelescoping plastic that can be adjusted to let the QB stand astall as 175 centimeters or as short as 81 centimeters.4. Social and demographic changes are leading to a greaterdemand for housing. People are living longer, and choosing to marry later, and in recent years there has been a rise in the number of single-parent families. Added to this, the UK is experiencing immigration from other countries, e.g. from Poland[1]which has recently joined the EU. The result is an ever-larger number of smaller households, all requiring accommodation.Part III. Translate the following sentences into Chinese (20%)1. The problem in the paradox, he reasoned, is that we are confusinga description of sets of numbers with a description of sets of setsof numbers.2. That’s not becaus e Facebook is hell-bent on stripping awayprivacy protections, but because the popularity of Facebook and other social networking sites has promoted the sharing of allthings personal, dissolving the line that separates the privatefrom the public.3. The two-wheel—as opposed to a tricycle or quad—design makesit more maneuverable in tight spaces and helps keep its weight down to about 16 kilograms.4. The straw bales, it turns out, are all packed tightly inside a series ofprefabricated rectangular wooden wall frames, which are thenlime-rendered, dried and finally slotted together like giant Lego pieces, called ModCell panels.5. Although the UK is an urban society, more and more peopleare choosing to live on the edge of urban areas - with many relocating to the countryside. This is called counter-urbanisation.。
Control Theory-- History, Mathematical Achievements and Perspectives
1.
Introduction
This article is devoted to present some of the mathematical milestones of Control Theory. We will focus on systems described in terms of ordinary differential equations. The control of (deterministic and stochastic) partial differential equations remains out of our scope. However, it must be underlined that most ideas, methods and results presented here do extend to this more general setting, which leads to very important technical developments. The underlying idea that motivated this article is that Control Theory is certainly, at present, one of the most interdisciplinary areas of research. Control Theory arises in most modern applications. The same could be said about the very first technological discoveries of the industrial revolution. On the other hand, Control Theory has been a discipline where many mathematical ideas and methods have melt to produce a new body of important Mathematics. Accordingly, it is nowadays a rich crossing point of Engineering and Mathematics. Along this paper, we have tried to avoid unnecessary technical difficulties, to make the text accessible to a large class of readers. However, in order to introduce some of the main achievements in Control Theory, a minimal body of basic mathematical concepts and results is needed. We develop this material to make the text self-contained. These notes contain information not only on the main mathematical results in Control Theory, but also about its origins, history and the way applications and interactions of Control Theory with other Sciences and Technologies have conducted the development of the discipline. The plan of the paper is the following. Section 2 is concerned with the origins and most basic concepts. In Section 3 we study a simple but very interesting example: the pendulum. As we shall see, an elementary analysis of this simple but important mechanical system indicates that the fundamental ideas of Control Theory are extremely meaningful from a physical viewpoint. In Section 4 we describe some relevant historical facts and also some important contemporary applications. There, it will be shown that Control Theory is in fact an interdisciplinary subject that has been strongly involved in the development of the contemporary society. In Section 5 we describe the two main approaches that allow to give rigorous formulations of control problems: controllability and optimal control. We also discuss their mutual relations, advantages and drawbacks. In Sections 6 and 7 we present some basic results on the controllability of linear and nonlinear finite dimensional systems. In particular, we revisit the Kalman approach to the controllability of linear systems, and we recall the use of Lie brackets in the control of nonlinear systems, discussing a simple example
Inside-Out Bulge Formation and the Origin of the Hubble Sequence
a rXiv:as tr o-ph/991423v129Jan1999Inside-Out Bulge Formation and the Origin of the Hubble Sequence By Frank C.van den Bosch Department of Astronomy,University of Washington,Box 351580,Seattle,WA 98195,USA Galactic disks are thought to originate from the cooling of baryonic material inside virialized dark halos.In order for these disks to have scalelengths comparable to observed galaxies,the specific angular momentum of the baryons has to be largely conserved.Because of the spread in angular momenta of dark halos,a significant fraction of disks are expected to be too small for them to be stable,even if no angular momentum is lost.Here it is suggested that a self-regulating mechanism is at work,transforming part of the baryonic material into a bulge,such that the remainder of the baryons can settle in a stable disk component.This inside-out bulge formation scenario is coupled to the Fall &Efstathiou theory of disk formation to search for the parameters and physical processes that determine the disk-to-bulge ratio,and therefore explain to a large extent the origin of the Hubble sequence.The Tully-Fisher relation is used to normalize the fraction of baryons that forms the galaxy,and two different scenarios are investigated for how this baryonic material is accumulated in the center of the dark halo.This simple galaxy formation scenario can account for both spirals and S0s,but fails to incorporate more bulge dominated systems.2 F.C.van den Bosch:Inside-Out Bulge Formationclumps that form stars,and this clumpiness is likely to result in a bulge.Even if the low-angular momentum material accumulates in a disk,the self-gravity of such a small, compact disk makes it violently unstable,and transforms it into a bar.Bars are efficient in transporting gas inwards,and can cause vertical heating by means of a collective bending instability.Both these processes lead ultimately to the dissolution of the bar;first the bar takes a hotter,triaxial shape,but is later transformed in a spheroidal bulge component.There is thus a natural tendency for the inner,low angular momentum baryonic material to form a bulge component rather than a disk.Because of the ongoing virialization,subsequent shells of material cool and try to settle into a disk structure at a radius determined by their angular momentum.If the resulting disk is unstable,part of the material is transformed into bulge material.This process of disk-bulge formation is self-regulating in that the bulge grows until it is massive enough to sustain the remaining gas in the form of a stable disk.I explore this inside-out bulge formation scenario,by incorporating it into the standard Fall&Efstathiou theory for disk formation.The ansatz for the models are the properties of dark halos,which are assumed to follow the universal density profiles proposed by Navarro,Frenk&White(1997),and whose halo spin parameters,λ,follow a log-normal distribution in concordance with both numerical and analytical studies.I assume that only a certain fraction,ǫgf,of the available baryons in a given halo ultimately settles in the disk-bulge system.Two extreme scenarios for this galaxy formation(in)efficiency are considered.In thefirst scenario,which I call the ‘cooling’-scenario,only the inner fractionǫgf of the baryonic mass is able to cool and form the disk-bulge system:the outer parts of the halo,where the density is lowest,but which contain the largest fraction of the total angular momentum,never gets to cool.In the second scenario,referred to hereafter as the‘feedback’-scenario,the processes related to feedback and star formation are assumed to yield equal probabilities,ǫgf,for each baryon in the dark halo,independent of its initial radius or specific angular momentum,to ultimately end up in the disk-bulge system.The values ofǫgf are normalized byfitting the model disks to the zero-point of the observed Tully-Fisher relation.Recent observations of high redshift spirals suggest that the zero-point of the Tully-Fisher relation does not evolve with redshift.This implies that the galaxy formation efficiency,ǫgf,was higher at higher redshifts(see vdb98for details).Disks are modeled as exponentials with a scalelength proportional toλtimes the virial radius of the halo(as in the disk-formation scenario of Fall&Efstathiou).The bulge mass is determined by requiring that the disk is stable.Since the amount of self-gravity of the disk is directly related to the amount of angular momentum of the gas,the disk-to-bulge ratio in this scenario is mainly determined by the spin parameter of the dark halo out of which the galaxy forms.3.Clues to the formation of bulge-disk systemsConstraints on the formation scenario envisioned above can be obtained from a com-parison of these disk-bulge-halo models with real galaxies.From the literature I compiled a list of∼200disk-bulge systems,including a wide variety of galaxies:both high and low surface brightness spirals(HSB and LSB respectively),S0,and disky ellipticals(see vdB98for details).After choosing a cosmology and a formation redshift,z,I calculate, for each galaxy in this sample,the spin parameterλof the dark halo which,for the assumptions underlying the formation scenario proposed here,yields the observed disk properties(scale-length and central surface brightness).We thus use the formation sce-nario to link the disk properties to those of the dark halo,and use the known statistical properties of dark halos to discriminate between different cosmogonies.The main results are shown in Figure1,where I plot the inferred values ofλversusF.C.van den Bosch:Inside-Out Bulge Formation3Figure1.Results for a OCDM cosmology withΩ0=0.3.Plotted are the logarithm of the spin parameter versus the logarithm of the disk-to-bulge ratio.Solid circles correspond to disky ellipticals,stars to S0s,open circles to HSB spirals,and triangles to LSB spirals.The thick solid line is the stability margin;halos below this line result in unstable disks.As can be seen,real disks avoid this region,but stay relatively close to the stability margin,in agreement with the self-regulating bulge formation scenario proposed here.The dashed curves correspond to the1, 10,50,90,and99percent levels of the cumulative distribution of the spin parameter.Upper panels correspond to the cooling scenario,and lower panels to the feedback scenario.Panels on the left correspond to z=0,middle panels to z=1,and panels on the right to z=3.the observed disk-to-bulge ratio for the galaxies in the sample.The dotted lines outline the distribution function of halo spin parameters of dark halos;it can thus be inferred what the predicted distribution of disk-to-bulge ratios is for galaxies that form at a given formation redshift.Results are presented for an open cold dark matter(OCDM) model withΩ0=0.3and no cosmological constant(ΩΛ=0).These results are virtually independent of the value ofΩΛ,but depend strongly onΩ0,which sets the baryon mass fraction of the Universe.Throughout,a universal baryon density ofΩb=0.0125h−2is assumed,in agreement with nucleosynthesis constraints.The inferred spin parameters are larger for higher values of the assumed formation redshifts.This owes to the fact that halos that virialize at higher redshifts are denser.Since the scalelength of the disk is proportional toλtimes the virial radius of the halo,higher formation redshifts imply larger spin parameters in order to yield the observed disk scalelength.In the cooling scenario,the probability that a certain halo yields a system with a large disk-to-bulge ratio(e.g.,a spiral)is rather small.This is due to the fact that in this scenario most of the high angular momentum material never gets to cool to become part of the disk. The large observed fraction of spirals in thefield,renders this scenario improbable.For the feedback cosmogony,however,a more promising scenario unfolds:At high redshifts (z∼>1)the majority of halos yields systems with relatively small disks(e.g.,S0s),whereas systems that form more recently are more disk-dominated(e.g.,spirals).This difference owes to two effects.First of all,halos at higher redshifts are denser,and secondly,the redshift independence of the Tully-Fisher relation implies thatǫgf was higher at higher redshifts.Coupled to the notion that proto-galaxies that collapse at high redshifts are4 F.C.van den Bosch:Inside-Out Bulge Formationpreferentially found in overdense regions such as clusters,this scenario thus automatically yields a morphology-density relation,in which S0s are predominantly formed in clusters of galaxies,whereas spirals are more confined to thefield.4.Conclusions•Inside-out bulge formation is a natural by-product of the Fall&Efstathiou theory for disk formation.•Disk-bulge systems do not have bulges that are significantly more massive than re-quired by stability of the disk component.This suggests a coupling between the formation of disk and bulge,and is consistent with the self-regulating,inside-out bulge formation scenario proposed here.•A comparison of the angular momenta of dark halos and spirals suggests that the baryonic material that builds the disk can not loose a significant fraction of its angular momentum.This rules against the‘cooling scenario’envisioned here,in which most of the angular momentum remains in the baryonic material in the outer parts of the halo that never gets to cool.•If we live in a low-density Universe(Ω0∼<0.3),the only efficient way to make spiral galaxies is by assuring that only a relatively small fraction of the available baryons make it into the galaxy,and furthermore that the probability that a certain baryon becomes a constituent of thefinal galaxy has to be independent of its specific angular momentum, as described by the‘feedback scenario’.•If more extended observations confirm that the zero-point of the Tully-Fisher rela-tion is independent of redshift,it implies that the galaxy formation efficiency,ǫgf,was higher at earlier times.Coupled with the notion that density perturbations that collapse early are preferentially found in high density environments such as clusters,the scenario presented here then automatically predicts a morphology-density relation in which S0s are most likely to be found in clusters.•A reasonable variation in formation redshift and halo angular momentum can yield approximately one order of magnitude variation in disk-to-bulge ratio,and the simple formation scenario proposed here can account for both spirals and S0s.However,disky ellipticals have too large bulges and too small disks to be incorporated in this scenario. Apparently,their formation and/or evolution has seen some processes that caused the baryons to loose a significant amount of their angular momentum.Merging and impulsive collisions(e.g.,galaxy harassment)are likely to play a major role for these systems.It thus seems that both‘nature’and‘nurture’are accountable for the formation of spheroids,and that the Hubble sequence has a hybrid origin.Support for this work was provided by NASA through Hubble Fellowship grant# HF-01102.11-97.A awarded by the Space Telescope Science Institute,which is operated by AURA for NASA under contract NAS5-26555.REFERENCESF all,S.M.,&Efstathiou,G.1980Formation and rotation of disc galaxies with haloesMNRAS193,189-206Navarro,J.F.,Frenk,C.S.,&White,S.D.M.1997A universal density profile from hierarchical clustering ApJ490,493-508van den Bosch,F.C.1998The formation of disk-bulge-halo systems and the origin of the Hubble sequence ApJ507,601-614。
Decay Constants $f_{D_s^}$ and $f_{D_s}$ from ${bar{B}}^0to D^+ l^- {bar{nu}}$ and ${bar{B}
form factor.
PACS index : 12.15.-y, 13.20.-v, 13.25.Hw, 14.40.Nd, 14.65.Fy Keywards : Factorization, Non-leptonic Decays, Decay Constant, Penguin Effects
∗ experimentally from leptonic B and Ds decays. For instance, determine fB , fBs fDs and fDs
+ the decay rate for Ds is given by [1]
+ Γ(Ds
m2 G2 2 2 l 1 − m M → ℓ ν ) = F fD D s 2 8π s ℓ MD s
1/2
(4)
.
(5)
In the zero lepton-mass limit, 0 ≤ q 2 ≤ (mB − mD )2 .
2
For the q 2 dependence of the form factors, Wirbel et al. [8] assumed a simple pole formula for both F1 (q 2 ) and F0 (q 2 ) (we designate this scenario ’pole/pole’): q2 F1 (q ) = F1 (0) /(1 − 2 ), mF1
∗ amount to about 11 % for B → DDs and 5 % for B → DDs , which have been mentioned in
Isospin constraints on the $tauto Kbar Knpinu$ decay mode
a r X i v :h e p -p h /9708473v 2 7 O c t 1997Isospin constraints on the τ→K ¯Knπνdecay modeAndr´e Roug´e ∗LPNHE Ecole Polytechnique-IN2P3/CNRSF-91128Palaiseau CedexAugust 1997AbstractThe construction of the complete isospin relations and inequalities between thepossible charge configurations of a τ→K ¯Knπνdecay mode is presented.Detailedapplications to the cases of two and three pions are given.X-LPNHE 97/081IntroductionThe isospin constraints on the τhadronic decay modes are known for the nπand Knπmodes[1,2].For the K ¯Knπfinal states,only the simplest decay mode K ¯Kπhas been ing the formalism of symmetry classes introduced by Pais[3],we generalize the relations for an arbitrary value of n and give a geometrical representation of the constraints.2The general methodThe K ¯Knπsystem produced by a τdecay has isospin 1;the possible values of the K ¯Kisospin I K ¯K are 0and 1and the isospin I nπof the n pion system is 1for I K ¯K =0and 0,1or 2for I K ¯K =1.Since there is no second-class current in the Standard Model,interferences between amplitudes with I K ¯K =0and I K ¯K =1vanish in the partial widths [2].Therefore we have the relationΓK 0¯K 0(nπ)−=ΓK +K −(nπ)−,(1)which is true for each charge configuration of the nπsystem,and,since ΓK S K S (nπ)−=ΓK L K L (nπ)−,ΓK +K −(nπ)−=ΓK S K L (nπ)−+2ΓK S K S (nπ)−,(2)using the most easily observable states.The amplitudes are classified by the values of I K¯K and I nπ.To complete the classifi-cation,we use the isospin symmetry class[3]of the nπsystem i.e.the representation of the permutation group S n to which belongs the state.It is characterized by the lengths of the three rows of its Young diagram(n1n2n3).Due to the Pauli principle,the momentum and isospin states have the same symmetry.Thus integration over the momenta kills the interferences between amplitudes in different classes and there is no contribution from them in the partial widths.Since I nπ=0and I nπ=1amplitudes belong to different symmetry classes[3],their in-terferences vanish.The presence of I nπ=2amplitudes makes the problem more intricate since they share symmetry properties with some I nπ=1or I nπ=0amplitudes[4,5].For instance,in the case n=2the symmetry class(200)is shared by I nπ=0and I nπ=2; in the case n=3,the symmetry class(210)is shared by I nπ=1and I nπ=2.Therefore the allowed domains in the space of the charge configuration fractions(f cc=Γcc/ΓK¯Knπ) must be determined separately for each symmetry class and I K¯K,taking interferences into account when necessary.The complete allowed domain is the convex hull of the sub-domains corresponding to the different I K¯K and symmetry classes and its projections are the convex hulls of their projections.For n≤5,which is always true in aτdecay,the isospin values and the symmetry class characterize unambiguously the amplitude properties[4,5],therefore there is,at most, one interference term per class.The sub-domain,for such a symmetry class associated with two different I nπvalues,is then a two-dimensional one since the partial widthsΓcc are linear functions of three quantities:the sums of squared amplitudes for the two values of I nπand the interference term.Its boundary is determined by the Schwarz’s inequality [6].This boundary is an ellipse;it can be parametrized by writing the sums of squared amplitudes for the two values of I nπasρ[1±cosθ]/2and the largest interference term allowed by the Schwarz’s inequality as kρsinθ,where the coefficient k depends on the coupling coefficients.The most general domain is hence the convex hull of a set of points corresponding to the symmetry classes without I nπ=2and a set of ellipses.The cases n=2and n=3are presented in detail in the following sections.They both have the property that only one symmetry class is associated with two isospin values. Higher values of n are not expected,for some time,to be of experimental interest.3The decayτ→K¯K2πνThe possible states that can be observed for aτ→νK¯Kππdecay are the following:K S K Sπ0π−K S K Lπ0π−K L K Lπ0π−K+K−π0π−K+¯K0π−π−K0K−π+π−K0K−π0π0.As mentioned before,not all the corresponding partial widths are independent and we can use the four fractions:2f K+K−π0π−=f K+K−π0π−+f K0¯K0π0π−,f K+¯K0π−π−,f K0K−π+π−and f K0K−π0π0,whose sum is equal to1,to describe the possible charge configurations in a three-dimensional space.The ratio K S K S/K S K L is a free parameter independent of the charge configuration fractions.The partial widths for all the charge configurations can be expressed as functions ofthe positive quantities S[IK¯K ,Iππ]which are the sums of the squared absolute values ofthe amplitudes with the given values of the isospins and the interference term I of the Iππ=0and Iππ=2amplitudes.With only two pions the coefficients are readily obtained from a Clebsch-Gordan table and we getΓK+K−π0π−+ΓK0¯K0π0π−=2ΓK+K−π0π−=S[0,1]+110S[1,2]ΓK+¯K0π−π−=63S[1,0]+130S[1,2]+IΓK0K−π0π0=130S[1,2]−I.The partial widthΓK¯Kππis the sum over the charge configurations:ΓK¯Kππ= ccΓcc= sc S sc.(4) The Schwarz’s inequality bounding the interference term I is|I|≤254(f K0K−π+π−−2f K0K−π0π0+16f K+¯K0π−π−).As explained before,the complete domain is the convex hull of this ellipse and the two points[0,1]and[1,1].Calling I the point of the ellipse for which f K0K−π0π0=0,the domain is made of the tetrahedron having the points I,[0,1],[1,1]and[1,0]for vertices and of the two half-cones whose bases are the halves of the ellipse delimited by the points I and[1,0]and whose vertices are the points[1,1]and[0,1]respectively.0.512f(K -K +π-π0)f (K 0K -π0π0)+f (K 0K -π+π-)0,1]0.512f(K -K +π-π0)f (K 0K +π-π-)+f (K 0K -π+π-)0,1]Figure 1:Projections of the allowed domain on the planes x/y ,x =2f K +K −π0π−,y =f K 0K −π+π−+f K 0K −π0π0and x =2f K +K −π0π−,y =f K +¯K 0π−π−+f K 0K −π+π−.The classes of amplitudes are labelled by the isospin values,[I K ¯K ,I 2π].For practical purposes it is useful to draw the projections of the domain on two-dimensional planes.The method is very simple:we first draw the projection of the ellipse on the plane and then the tangents to the projected ellipse from the projections of the points [0,1]and [1,1].A first simple example is the projection on the plane x/y ,with x =2f K +K −π0π−and y =f K 0K −π+π−+f K 0K −π0π0.Here the ellipse projection is a mere segment and the projected domain is the polygon whose vertices are the (projected)points [0,1],[1,1],[1,0]and [1,2].More interesting is the projection x =2f K +K −π0π−,y =f K +¯K 0π−π−+f K 0K −π+π−,since the two final states K +¯K0π−π−and K 0K −π+π−have the same topology:one K 0and three charged hadrons.The complement 1−x −y is the fraction f K 0K −π0π0of decays with two π0’s.The projected ellipse has vertical tangents at the points [1,0]and [1,2].It is also tangent to the line x +y =1at the point I (x =1/4),for which W [1,0]=W [1,2]/5since f K 0K −π0π0can be 0,because of the interference,only when the two contributions have the same modulus.The second tangent from [0,1]touch the ellipse at the point of coordinates x =2/23and y =12/23.The allowed domain is shown on Fig.1.The main constraint is the inequalityf K 0K −π0π0≤3implies the dominance of I K¯K=1and a small value the dominance of I K¯K=0.With one dominant isospin for the K¯K system,the ratio K S K S/K S K L measures the proportions of the two G-parities i.e.the contributions of axial and vector currents.4The decayτ→K¯K3πνThefinal states for the decayτ→K¯Kπππνare:K0¯K0π+π−π−K+K−π+π−π−K0¯K0π0π0π−K+K−π0π0π−K0K−π+π−π0K0K−π0π0π0K+¯K0π−π−π0.The relations between the K0¯K0and K+K−final states are the same as in the τ→K¯Kππνdecay.Thus the charge configurations are described in a four-dimensional space by thefive fractions:2f K+K−π+π−π−=f K+K−π+π−π−+f K0¯K0π+π−π−,2f K+K−π0π0π−= f K+K−π0π0π−+f K0¯K0π0π0π−,f K0K−π+π−π0,f K0K−π0π0π0and f K+¯K0π−π−π0.We shall label the amplitudes by the two isospin values and the symmetry class: [I K¯K,(n1n2n3)I3π].The relations between the partial widths for the charge configurations and the amplitudes use both Clebsch-Gordan coefficients and the similar coefficients for the symmetry classes[4,5].With the notations defined in the previous section,they can be written2ΓK+K−π−π0π0=ΓK0¯K0π−π0π0+ΓK+K−π−π0π0=12S[0,(210)1]+14S[1,(210)1]+35S[0,(300)1]+15S[1,(300)1]+120S[1,(210)2]−IΓK0K−π+π−π0=S[1,(111)0]+12S[1,(210)1]+1√10S[1,(300)1](8)ΓK+¯K0π−π−π0=3√25S[1,(210)1]S[1,(210)2].(11)2f(K +K -π-π-π+)f (K 0K +π-π-π0)+f (K 0K -π+π-π0)0Figure 2:Projection of the allowed domain on the plane x/y ,x =2f K +K −π+π−π−,y =f K 0K −π+π−π0+f K +¯K 0π−π−π0.The classes of amplitudes are labelledby the isospin values and symmetry classes,[I K ¯K ,(n 1n 2n 3)I3π].The sub-domains are points for the classes [0,(300)1],[0,(210)1],[1,(300)1]and [1,(111)0].For the interfering classes [1,(210)1]and [1,(210)2],the plane of the two-dimensional sub-domain is determined by the two relations:ΓK 0K −π0π0π0=0and 2(1+13)ΓK +K −π−π0π0+2(1−13)ΓK +K −π+π−π−−ΓK 0K −π+π−π0−1To distinguish twoπ0from threeπ0decays,we can use a third coordinate z= f K0K−π0π0π0.The three-dimensional domain is the cone having for basis the above de-scribed contour in the x/y plane and,for vertex,the point[1,(300)1].5SummaryWe have presented the complete isospin constraints on theτ→K¯Knπνdecay modes in the space of charge configurations with some details in the cases n=2and n=3.The geometrical method adopted allows to draw very easily any wanted projection of the multi-dimensional domain and hence obtain the most restrictive inequalities for a given set of measurements.References[1]F.J.Gilman and S.H.Rhie,Phys.Rev.D31(1985)1066[2]A.Roug´e,Z.Phys.C70(1996)109[3]A.Pais,Ann.Phys.9(1960)548[4]A.Pais,Ann.Phys.22(1963)274[5]H.Pilkuhn,Nucl.Phys.22(1961)168[6]L.Michel,Nuovo Cim.22(1961)203。
Constraints on the Charged Higgs Sector from B Physics
ANL-HEP-CP-92-125 November 1992
CONSTRAINTS ON THE CHARGED HIGGS SECTOR FROM B PHYSICS ∗
JOANNE L. HEWETT High Energy Physics Division Argonne National Laboratory
Many extensions of the Standard Model (SM) predict the existence of an enlarged Higgs sector beyond the minimal one-doublet version[1]. The simplest extensions are models with twoHiggs-Doublets (2HDM), which predict a physical spectrum of three neutral Higgs scalars, two of which are CP-even (h0, H0) while one is CP-odd (A0), and two charged Higgs scalars (H±). We consider two distinct 2HDM which naturally avoid tree-level flavor changing neutral currents. In Model I, one doublet (φ2) provides masses for all fermions and the other doublet (φ1) decouples from the fermion sector. In a second model (Model II), φ2 gives mass to the up-type quarks, while the down-type quarks and charged leptons receive their masses from φ1. Supersymmetry and many axion theories predict couplings of the type present in Model II. Each doublet obtains a vacuum expectation value (vev) vi, subject only to the constraint that v12 + v22 = v2, where v is the usual vev present in the SM. In a general 2HDM, the charged Higgs mass mH± and the ratio of vevs, v2/v1 ≡ tan β, are a priori free parameters, as are the masses of all the neutral Higgs fields. However, in supersymmetric models, mass relationships exist between the various Higgs scalars. At tree-level, in such models, only two parameters are required to fix the masses and couplings of the entire scalar sector, but once radiative corrections are included[2], the values of the top-quark and squark masses also need to be specified.
TIMIT
5 Output from T¯ I MIT 5.1 Hydrostatic Quantities . . . . . . . . . . . . . . . 5.1.1 The format of the hydrostatic output . . 5.2 Time Domain Hydrodynamic Quantities . . . . . 5.2.1 Format of the time domain hydrodynamic 5.3 Frequency Domain Hydrodynamic Quantities . . 5.3.1 Format of the frequency domain output .
T¯ I MIT
A panel-method program for transient wave-body interactions.
VERSION 4.0: For zero and forward speed analysis of a single body with any number of waterlines, arbitrary wave heading, generalized modes, and infinite or finite depth.
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The development of T¯ I MIT has been supported by the Office of Naval Reseach, the Joint Industry Project “Wave Effects on Offshore Structures”, the Consortium for Numerical Analysis of Wave Effects on Offshore Structures, and the Naval Ship Warfare Center.
Constraints on a General Higgs Sector from $K^0-bar K^0$, $B_d- bar B_d$ Mixing and the $ep
Abstract The scalar sector of the Standard Model is extended to include an arbitrary assortment of scalars. In the case where this assignment does not preserve ρ = 1 at the tree-level, the departure from unity itself puts the most stringent constraint on the scalar sector, and where ρtree = 1 is maintained, useful bounds on the parameter space of the charged Higgs mass and the doublet-nondoublet mixing angle can arise ¯d , K 0 − K ¯ 0 mixing and the ǫ parameter. These from data on Bd − B constraints turn out to be comparable (and in some cases, better) to those obtained from Zail: rupa@tnp.saha.ernet.in E-mail: akundu@saha.ernet.in
The electroweak symmetry breaking sector of the Standard Model (SM) is still as cloudy as it was in the time of its formulation; and the main factor responsible for this is the absence of any direct evidence of the Higgs boson. The minimal version of the SM requires one complex scalar doublet to break the electroweak symmetry; however, there is no a priori reason why more scalars cannot exist. Models with two or more doublets have been explored in this spirit [1]. It is also pertinent to investigate the consequences of scalars belonging to non-doublet representations of SU (2). This will enlarge the particle content of the SM, and change the gauge-scalar as well as the fermion-scalar interactions, without affecting the SU (2)L × U (1)Y gauge structure of the model. That these non-doublet scalar representations can induce Majorana masses for left-handed neutrinos has been shown [2]. Collider signatures of scalars belonging to a triplet representation have also been investigated [3]. However, there is one serious constraint on these higher dimensional (> 2) scalar representations: they in general do not maintain ρ = 1 at tree-level. Singlet and doublet representations do not suffer from this malady and that is why much work have been done on their phenomenological implications [4, 5]. For an arbitrary assortment of scalars, one has three possibilities: 1. The higher dimensional multiplet does not incidentally contribute to ρ. This will happen, e.g., for a multiplet with weak isospin T = 3 and weak hypercharge Y = 4. However, being quite artificial, such representations will not be discussed anymore in this paper. 2. The vacuum expectation values (VEV) of the higher representations are much smaller than the doublet VEV so that ρ − 1 is within experimental bound. 3. There is a remaining custodial SU (2) symmetry among the higher representations. In this case, the effects of the ‘bad’ representations on ρ − 1 cancel out. For such a cancellation to remain valid even at oneloop level, one requires a fine-tuning; however, it has been shown [5] that the fine-tuning required is of the same order as one encounters in 1
Constraints and universal algebra
In a previous paper 5] we explored one example of this by describing and using the close links between constraint satisfaction problems and relational databases. Relational database theory provides a very powerful and convenient terminology for describing operations on relations, and certain aspects of their structure, and this terminology is very useful in the description and study of constraint satisfaction problems. Furthermore, many of the computational tasks undertaken in the processing and solution of constraint satisfaction problems also arise in the context of database management. We believe that a greater interchange of ideas between these two elds could be of considerable bene t to both 2, 5], and therefore these links should receive more attention than they currently do.
Zitterbewegung Modeling
In:Found.Physics.,Vol.23,No.3,(1993)365–387. Zitterbewegung ModelingDavid HestenesAbstract.Guidelines for constructing point particle models of the elec-tron with zitterbewegung and other features of the Dirac theory are dis-cussed.Such models may at least be useful approximations to the Diractheory,but the more exciting possibility is that this approach may lead toa more fundamental reality.1.INTRODUCTIONFor many years I mulled over a variation of Schr¨o dinger’s zitterbewegung concept to account for some of the most perplexing features of quantum theory.I was reluctant to publish my ideas,however,because the supporting arguments were mainly qualitative,and physics tradition demands a quantitative formulation which can be subjected to experi-mental test.Unfortunately,the road to a quantitative version appeared to be too long and difficult to traverse without help from my colleagues.But how could I enlist their help without publishing my qualitative arguments to get their attention?The impetus to break this impasse came from the inspring example of Asim Barut. Barut is unusual among theoretical physicists,not only for the rich variety of clever ideas he generates,but also for publishing theoretical fragments which are sometimes mutually incompatible.Though each of his papers is internally coherent,he is not afraid to contradict himself from one paper to the next or try out alternative theoretical styles.He rapidly explores one idea after another in print,searching,I suppose,for ever deeper insights. Consequently,his papers are an exceptionally rich source of new ideas,arguments,and viewpoints.It was Barut’s inspiring example that convinced mefinally to publish my qualitative analysis of the zitterbewegung in Ref.1.Despite its defects,that publication did,indeed,stimulate helpful interaction with my peers and led to a mathematically well-grounded zitterbewegung interpretation of the Dirac theory[2]which is the starting point for the present paper.In appreciation for his positive influence on these events,I dedicate this paper to Asim Barut.The main motivation for analyzing zitterbewegung(ZBW)models is to explain the elec-tron’s spin S and magnetic momentµas generated by a local circulation of mass and charge. Experimental evidence rules out the possibility that the electron is an extended body,for the relativistic limitation on velocity implies that,to generate S andµ,the dimensions of the body cannot be less than a Compton wavelength(≡10−13m),whereas scattering experiments show that the electron cannot be larger than10−16m.This leaves open the possibility that the electron can be regarded as a point charge which generates S andµby an inherent local circular motion.For a particle moving in a circle at the speed of light c,the radius of the circle¯λis related to the circular frequencyωby¯λω=c=1(1.1)1If the particle is presumed to have mass m and generate the electron spin by this motion, then|S|=12¯h=mωλ2(1.2) Thus,radius(wavelength),mass,and frequency are all related by¯λ=¯h2m=1ω(1.3)This may appear to be a hopelessly simple-minded explanation for electron spin,but(sur-prise!!)it has been shown to be completely consistent with the mathematical structure of the Dirac theory.(2)Thus,despite the appearance of electron mass in the Dirac equation, the electron may indeed be moving with the speed of light.Contrary to long-standing “classical”arguments,such motion can produce a gyromagnetic ratio of2.Moreover,the complex phase factor can be interpreted as a direct representation of the circular ZBW. Thus,the phase factore−iϕ/¯h=e−iθ/2(1.4) is associated with the frequencyω=˙θ,where differentiation is with respect to the proper time on the worldline of the center of curvature for the circular ZBW.Note that(1.4)is a half-angle representation of the parison of(1.4)with(1.3)gives˙ϕ=12¯hω=m(1.5)showing that m must be a variable mass.It is equal to the electron rest mass m0only for a free particle.This interpretation of the phaseϕis derived from the standard energy-momentum operatorpµ=i¯h∂µ−eAµ(1.6) Operating on the phase factor(1.4),we obtain the gauge-invariant energy-momentum vectorpµ=∂µϕ−eAµ(1.7)Contraction with the“Dirac velocity”vµ(for the center of curvature)givesp·v=˙ϕ−eA·v(1.8)For m=p·v this differs from(1.5),an important point to be discussed later.The compatibility of this ZBW interpretation with the details of the Dirac theory is demonstrated in Ref.2.Its very success,however,suggests the possibility of a deeper theory of electrons.It suggest that the Dirac theory actually describes a statistical ensemble of possible electron motions.If that is correct,it should be possible tofind equations of motion for a single electron history and then derive the Dirac theory by statistical arguments.That possibility is explored in this paper,where definite equations of motion for a single point particle with ZBW are written down and discussed.With the ZBW interpretation as a guide,the equations are designed to capture the essential features of the Dirac equation. Of course,such“working backward”from presumed statistical averages to underlying dynamics is a guessing game,prone to error in its early stages.For that reason,this paper makes no pretense of producing a definitive ZBW model.The objective,rather,is2(1)to motivate and describe both kinematical and dynamical aspects of the ZBW concept,(2)to formulate and implement basic design principles for ZBW modeling,and(3)to identify and sharpen issues that must be resolved to produce a fully viable theory.This is not the place to discuss experimental tests of the ZBW.If the resulting equations do not in fact describe a deeper reality,at least they may constitute a useful approximation to the Dirac theory,a new version of“semiclassical”approximations which have already proved their value.2.OBSER V ABLES OF THE DIRAC THEORYThe present paper is a continuation of work published in this journal,(2)so some famil-iarity with that work can be presumed as background.In particular,spacetime algebra will be employed as the essential mathematical tool.The rudiments of spacetime algebra are explained in Ref.2and expanded in many references cited therein.A basic feature of the algebra is that the fourγµin the Dirac theory are not regarded as matrices but as vectors constituting afixed orthonormal frame for spacetime.As a guide for the design of a ZBW model,the formulation of Dirac observables in terms of spacetime algebra is reviewed here.The Dirac wave function for a electron can be written in the formψ=(ρe iβ)1/2R(2.1) whereρandβare scalars,i=γ0γ1γ2γ3is the unit pseudoscalar,and R is a“unimodular”spinor,that is,an even multivector satisfyingRR =1(2.2) The form(2.1)can be interpreted physically as a factorization of the electron wave function into a statistical factor(ρe iβ)1/2and a kinematical factor R.Thus,the scalar modulusρis interpreted as a probability density,although the interpretation ofβraises problems to be discussed later.The kinematical factor R=R(x)determines an orthonormal frame of vectors eµ=eµ(x) at each spacetime point x byeµ=RγµR (2.3) This frame describes the kinematics of electron motion.The Dirac currentJ=ψγ0ψ=ρe0(2.4)has vanishing divergence,so it determines a family of electron streamlines with tangentvectorv=dxdτ=x.=e0(2.5)The vectorfield v=v(x)can be interpreted statistically as a prediction of the electron velocity at each spacetime point x.The electron spin(or polarization)vector is defined bys=12¯h e3(2.6)3However,the electron spin is more properly characterized as a bivector quantity S defined byS =12¯h e 2e 1=isv (2.7)The unimodularity condition (2.2)implies that the derivative of R in the direction γµcan be written in the form∂µR =12ΩµR(2.8)where Ωµis a bivector quantity.Consequently,the derivatives of (2.3)can be written∂µe ν=Ωµ·e ν(2.9)showing that Ωµis to be interpreted as the rotational velocity of the frame {e µ}when displaced in the γµdirection.A physical meaning is assigned to the Ωµby comparing them to the spin S .WriteΩµS =Ωµ·S +Ωµ×S +Ωµ∧S =P µ+iq µ+∂µS(2.10)whereP µ=P ·γµ=Ωµ·Sq µ=q ·γµ=−i (Ωµ∧S )=Ωµ·(−iS )=Ωµ·(sv )=v ·(∂µs )=−s ·(∂µv )∂µS =Ωµ×S (2.13)(2.11)(2.12)In the Dirac theory,the vector P in (2.11)is implicitly interpreted as the canonical (energy)momentum ,which is related to the kinetic (energy)momentum vector p byP =p +eA (2.14)where A =A µγµis the electromagnetic vector potential and e is the electron charge.Thus,Eq.(2.11)explicitly attributes the energy-momentum of the electron to the rotation rate in the plane of the spin S .This is a key to the ZBW interpretation,and identifies (2.11)as the full generalization of (1.7).Along an electron streamline,the rotational velocity isΩ=v µΩµ(2.15)where v µ=v ·γµ,so (2.8)yields ˙R =12ΩR (2.16)where the overdot indicates differentiation with respect to proper time along the streamline.The “motion”of the e µ,along a streamline is therefore described by˙e µ=Ω·e µ(2.17)In particular,˙v =Ω·v(2.18)4becomes an equation of motion for the streamline whenΩ=Ω(x)is specifled.And(2.13) yields˙S=Ω×S(2.19) which determines the spin precession along a streamline.Thus,the dynamics of electron motion can be reduced to determiningΩ,a matter to be discussed later.Along a streamline,(2.10)and(2.12)yieldΩS=Ω·S+Ω×S+Ω∧S=v·P+˙S+i(˙s·v)(2.20) This summarizes the relation ofΩto observables.Indeed,it can be solved forΩby multi-plication with S−1.Furthermore,(2.11)and(2.14)yieldm=p·v=Ω·S−eA·v(2.21) which defines a variable mass m for the electron in accordance with the remarks in the introduction.All the above relations are implicit in the standard Dirac theory and do not involve any approximation,so they are equally applicable to the physical interpretation of any solution of the Dirac equation.The zitterbewegung interpretation requires one further physical assumption which,however,does not alter the mathematical structure of the Dirac theory. To account for the spin S as generated by electron motion,the electron must have a component of velocity in the spin plane.This is readily achieved by identifying the electron velocity with the null vectoru=e0−e2(2.22) From(2.17)it follows that,for specifiedΩ=Ω(x),it has the equation of motion˙u=Ω·u(2.23) Its solutions are always lightlike helixes winding around the electron streamlines with ve-locity v=e0.3.ZITTERBEWEGUNG KINEMATICSNow we get to the main business of this article:to construct and analyze point par-ticle models of the electron consistent with the features of the Dirac theory described in the preceding section.Thefirst step is to define the appropriate kinematic variables and constraints.The electron is presumed to have a lightlike history z=z(τ)in spacetime with velocityu=dzdτ=˙z,so u2=0(3.1)A proper time cannot be defined on such a curve,so an additional condition is needed below to specify the parameterτuniquely.5The electron motion is assumed to possess an intrinsic angular momentum,or spin ,S =S (τ),a spacelike bivector with constant magnitude specified byS 2=−¯h 24(3.2)The velocity u can be decomposed into a component v orthogonal to S and a component ˙r in the S -plane defined byv =(u ∧S )S −1(3.3)˙r =(u ·S )S −1(3.4)Note that v and ˙r are merely auxiliary variables defined by these equations for purposes of analysis.This determines the decompositionu =v +˙r (3.5)withv ·˙r =0(3.6)and,by (3.1),v 2=−˙r 2(3.7)A scaling for the parameter τis determined by assuming thatv 2=v ·u =(u ∧S )2S 2=1(3.8)It will be seen later,though,that this is a mere convention,and not really a physical constraint on the model.Equation (3.5)suggests the decompositionz (τ)=x (τ)+r (τ)(3.9)with v =x .(3.10)This expresses the electron history z (τ)as a circulating motion with radius vector r (τ),angular momentum S ,and a center of curvature with a timelike history x =x (τ).All these features of our particle model correspond exactly to kinematic features of theDirac theory.Indeed,(3.5)is the same as (2.22)with v =e 0and ˙r=−e 2.It follows that the definition of a comoving frame by (2.3)and the spinor equation of motion (2.8),with their sundry implications (2.17),(2.18),(2.19),and (2.22),apply also to the particle model,the sole difference being that all quantities are defined only on the particle history z (τ)rather than as fields distributed over spacetime.As in the Dirac theory,the dynamics in those equations are entirely determined by specifying the rotational velocity Ω=Ω(z (τ)),a matter to be discussed in the next section.The situation is different,however,for momentum and mass.We define the momentum p in a standard way by writingp =mu =mv +m ˙r (3.11)6This cannot be identified with the momentum p defined in the Dirac theory by(2.11)and (2.14),because it has the rapidlyfluctuating component mr.Rather,it will be argued later that the Dirac momentum corresponds to the average of(3.11)over a ZBW period.The mass m in(3.11)remains to be defned.In accordance with the ZBW concept,we assume that it has the same origin as the spin,following(2.21)in defining it in relation to (3.11)bym=Ω·S=p·v=−p·˙r(3.12) This may differ from(2.21),however,due to averaging over a ZBW period.The essential kinematic specifications for our model are now complete,but it will be convenient to relate mass and momentum to spin by writingS=m˙r∧r(3.13)This amounts to a definition of the vector radius of curvature.Substitution into(3.12) yieldsΩ·(˙r∧r)=1(3.14) This generalizes Eq.(1.1),whereω=Ω·ˆS=2¯hΩ·S(3.15)is the ZBW(circular)frequency.The mass m,the scalar radius of curvature¯λ=|r|,and the ZBW frequencyωall covary with changes inΩ.To deduce how thay are interrelated,note that by(2.17),˙r=−e2 implies that..r=Ω·˙r(3.16)which,in turn,implies..r·˙r=0(3.17) Also,r2=−|r|2=¯λ2implies that˙r·r=−¯λ˙¯λ(3.18) From(3.13)and(2.19),we obtain˙S=˙m˙r∧r+m..r∧r=Ω×S=m(Ω·˙r)∧r+m˙r∧(Ω·r)Whence,˙m˙r∧r=m˙r∧(Ω·r)(3.19) Multiplying this by˙r∧r and using(˙r∧r)2=−¯h22m2(3.20)obtained from(3.2),wefind˙m−¯h24m2=m(˙r·r)(Ω·r)·˙r=m¯λ˙¯λ7Therefore,−¯h2 8˙mm=ddτ¯h24m=d¯λ2dτ(3.21)Setting the integration constant to zero,we thus obtainm¯λ=12¯h(3.22) exactly as specified by(1.3).Here,however,it is clear that m and¯λcan be variable,with derivatives related by˙m m =−˙¯λ¯λ(3.23)It should be evident that we have a new concept of mass here,through,to a certain extent,it was already implicit in the Dirac theory.The vague concept of mass as some kind of material stuffis completely gone.No longer is vanishing mass a distinguishing feature of particles moving with the speed of light,because p2=m2u2=0here.The relation m=p·v expresses the inertial property of mass as a relation between momentum and velocity.This relation conforms to the relativistic concept of mass as a measure of energy content,but that just means that mass and energy are essentially one andthe same.The complementary relation m=Ω·S=12¯hωasserts that mass is a frequencymeasure.This conforms to de Broglie’s original idea that the electron contains an internal clock with frequency determined by its mass,though for a free particle the ZBW frequency differs from the de Broglie frequency by a factor of2,and the mass varies with interactionsin the present model.Moreover,the relation m¯λ=12¯h says that this frequency measuresthe radius of curvature of the electron worldline,so it is a thoroughly geometrical quantity. All this suggests that the mass relates our externally imposed time scale to a time scale intrinsic to the electron.Considerations in the next section suggest that mass is a measure of self-interaction.In other words,mass measures the coupling strength of the electron to its ownfield.This, then,is the source of the inertial property of electron mass.Intuitively speaking,when accelerated by externalfields,the electron must drag its ownfield along.The direct relation of mass to ZBW frequency described here comes from assuming m=Ω·S=˙ϕ,which differs from the relations(2.21)and(1.8)in the Dirac theory by the eA·v term.This difference was introduced into the particle model for a physical reason. The Dirac equation allows the value of˙ϕto be changed by a gauge transformation,whereas the ZBW interpretation of˙ϕas an objective property of electron motion related to its spin implies that˙ϕshould have a unique value.It will be argued later that the eA·v term may appear in the Dirac theory from time averaging which is inherent in the theory.On the other hand,if the eA·v term is found to be essential to account for physical facts such as the Aharonov–Bohm effect,it could be incorporated into a particle model.This is a physical issue to be resolved by further theoretical and experimental analysis.4.ZITTERBEWEGUNG DYNAMICSAs already mentioned,the dynamics of electron motion is completely determined by specifying the rotational velocityΩ=Ω(z(τ)).An explicit expression forΩin the Dirac8theory was derived in Ref.3and is discussed in Section 6.With the Dirac Ωas a guide,our problem is to guess a suitable expression for Ωin our particle model.The simplest possibility to consider is Ω=e m 0F +m 0S −1(4.1)where m 0is the electron rest mass and the bivector F is any specified external electro-magnetic field.This determines a well-defined dynamical model which can account for a significant range of physical phenomena.Substituting (4.1)into (2.18)and using v ·S =0,we obtainm 0˙v =eF ·v (4.2)which is exactly the form of the classical Lorentz force,so it is evident that classical elec-trodynamics can be recovered from the model.However,if (4.2)is to be regarded as an equation of motion for the center of curvature x =x (τ)with v =x .,it is necessary to express F as a function of x instead of z .This is most naturally done by means of the first-order Taylor expansion.F (z )=F (x +r )=F (x )+r ·F (x )(4.3)where r ·=r µ∂µ,.Thus,the classical result will be obtained if and only if the last term in (4.3)in negligible compared to F (x ).Let us call this case the classical approximation .Of course,the last term in (4.3)is a ZBW effect.Substituting (4.1)into (2.19)yields˙S =e m 0F ×S (4.4)which exhibits the g =2value for the gyromagnetic ratio in the classical approximation,exactly as in the Dirac theory.Note that the ZBW correction on the right side of (4.3)vanishes when F is constant,so the classical approximation applies rigorously in that case,the case which has been of greatest interest in measurements of the g -factor.While (4.2)determines the electron’s center of curvature,and (4.4)determines the spin precession along the worldline,the remaining feature of electron motion is the ZBW fre-quency determined by substituting (4.1)into (3.12)or (3.15),with the resultm =e m 0F ·S +m 0(4.5)Thus,the external field produces a gauge-invariant mass shift of Larmor type.Instead of trying to integrate the equations for ˙v and ˙Sdirectly,it is advisable to employ the spinor equation (2.16)which,for Ωgiven by (4.1),becomes with the help of (2.2)and (2.7),˙R =e 2m 0F R +2m 0¯h Rγ2γ1(4.6)This equation has solutions of the formR =R 0e −γ2γ1ϕ/¯h (4.7)9where R 0satisfies the “reduced equation”˙R 0=e 2m 0F R 0(4.8)and the phase ϕcan be obtained by integrating (4.5)with m =˙ϕafter S has been found by integrating (4.8).Solutions of (4.8)in the classical approximation have been found in Ref.4for the cases where F is a constant,plane wave,or Coulomb field,and the equation x .=v was integrated to find explicit expressions for the worldline in each case.The form of (4.1)suggests various generalizations,such asΩ=e m 0(F +F s )(4.9)where F s is the electron’s self-field,that is,the electromagnetic field of the electron itself evaluated on the electron’s worldline.The self-field has the general form e m 0F s =m 0S −1+Ωex +e m 0F RR (4.10)where F RR is the radiative reaction field of the electron.Since radiative reaction is not included in the Dirac theory,we can defer discussion of the F RR term to another occasion.The remaining terms must then describe nonradiative effects of self-interaction.The term m 0S −1in (4.1)and (4.10),which is crucial for “generating”the electron rest mass and spin,has the bivector form of a magnetic field.This leads to the conclusion that the electron’s electromagnetic selfinteraction is fundamentally of magnetic type.This also raises the question of what form the electron’s field takes at points away from the electron worldline.A major reason for developing a ZBW model in the first place was to explain the electron’s magnetic dipole field.If this explanation is taken seriously,then the dipole field must be regarded as an average over a ZBW period,and the actual field must also contain a high-frequency component F zbw which oscillates with the ZBW frequency.As argued in Refs.1and 2,this has the potential for explaining some of the most prominent and perplexing features of quantum theory,such as electron diffraction and the Pauli principle,as resonant interactions mediated by the ZBW field F zbw .Note that (4.1)can be employed to study this possibility without further modification.It is only necessary to regard the external field Fin (4.1)as including ZBW fields from external sources.However,a quantitative analysis of ZBW resonances must be deferred to another time.Of course,the existence of a stable fluctuating ZBW field accompanying even a free elec-tron raises questions about radiation which must be addressed eventually.The viewpoint adopted here is that the Dirac theory suggests that such fields exist,and we should push the study of their putative effects on electron motion as far as possible before attacking the fundamental problems of radiation and self-interaction.Equation (4.10)describes the coupling of the electron to its own field.Even for a free particle,the m 0S −1term implies that the electron’s momentum (3.11)has a component mr which fluctuates (rotates )with the ZBW frequency.Presumably,this reflects a fluctuation of momentum in the electron’s ZBW field.The nonspecific term Ωex is included in (4.10)to cover the possibility that the10electron’sfield may have nonradiating excitations which appear even in the motion of a free particle.An example will be given in the next section.5.ZITTERBEWGUNG A VERAGESIt is of interest to eliminate ZBW oscillations from the equations of motion by averaging over a ZBW period for two good reasons at least.First,these high-frequency oscillations (on the order of1021Hz)are irrelevant in many situations(such as the classical limit),so it is advisable to systematically eliminate the need to consider them.Second,the(energy) momentum vector in the Dirac theory does not exhibit the ZBWfluctuations of the mo-mentum vector p defined by(3.11),so it must be regarded as some kind of average p of p. Thus,the study of ZBW averages is essential if the Dirac theory is to be derived from an underlying ZBW substructure.Even apart from the complication that the ZBW period T=2π/ωis a variable quantity, the definition of a ZBW average is not entirely straightforward,because it must be compati-ble with invariant constraints on ZBW kinematics.For example,for the center-of-curvature velocity the straightforward definition of an average v=v(τ)byv≡1Tτv(τ)dτ=x(τ)−x(0)T(5.1)is unsatisfactory,because it fails to preserve the constraint v2=1.For any curved historyx(τ),|x(τ)−x(0)|> τ|dx|=Timplies v2>1.This defect could be corrected by introducing a parameterβ0and revisingthe definition(5.1)tov cosβ0=1Tτv(τ)dτ(5.2)with v2=1by definition.A parameter likeβ0does indeed appear in the Dirac theory,but we shall entertain other,probably deeper,reasons for that.Since v is orthogonal to the ZBWfluctuations,it is affected by them only indirectly through the equations of motion. It appears permissible,therefore,to take v=v by definition.To maintain the constraint v·S=0,it is necessary to define S=S also.In Ref.2, I made the mistake of suggesting that v be defined by the ZBW average u.Instead,we take v as defined already by(3.3)and define p indirectly by imposing angular momentum conservation as a constraint.In Ref.2,it was correctly argued that compatibility with the Dirac theory requires that the average angular momentum J must satisfy the decomposition into orbital and spin components defined byJ=p∧z=p∧x+S(5.3) Of course,momentum conservation on the average requires that˙p=f(5.4)11where f is the average force on the particle.Angular momentum conservation then requires that˙J=˙p∧x+p∧v+˙S=f∧x(5.5)Therefore,˙S+p∧v=0(5.6) As noted in Ref.3,this equation wasfirst formulated by Wessenhoffand has been studied by many authors as a classical model for a particle with spin.Its appearance here gives it new significance as an approximation to the Dirac equation.Defining the average mass bym=p·v=p·v(5.7) and adding it to(5.6),we obtainvp=m+˙S(5.8) This can be solved explicitly forp=v(m+˙S)=mv+v·˙S=mv+S·˙v(5.9) where the last equality comes from differentiating v·S=0,and it is noted that(5.6) implies as well asv∧˙S=0(5.10) as well asp∧˙S=0(5.11) Consistency with the conservation laws thus implies that we must take(5.8)as the definition of p.It will be seen in the next section that this corresponds to the Gordon decomposition in the Dirac parison of(5.8)with(3.11)implies thatm˙r=v·˙S(5.12) an altogether reasonable result,which implies that p=mu=mv if and only if v·˙S=0. The mean dynamics is determined by assuming that it is governed by the averageΩof the rotational velocity over a ZBW period.This implies that˙S=Ω×S(5.13) for example.Accordingly,averaging(3.12)givesm=Ω·S=p·v(5.14) so the analog of(2.20)in this case takes the formΩS=m+˙S+i(˙s·v)(5.15) Whence,comparison with(5.8)givesΩS=vp+i(˙s·v)(5.16)12The last term in(5.16)is purely kinematical,but(5.10)puts a constraint on˙s,for the derivative of is=Sv=S∧v is then i˙s=S∧˙v=i(sv)·˙v.Whence,˙s=−(s·˙v)v=(˙s·v)v(5.17) which implies˙s·˙v=0(5.18) as well as˙v·˙S=0(5.19) An equation of motion for the mass can now be derived.From the right side of(5.9),it follows thatp·˙v=0(5.20) and(5.13)impliesΩ·˙S=Ω·(Ω×S)=0(5.21) for any value ofΩ.Therefore,the derivative of(5.14)gives˙m=˙Ω·S=v·˙p=v·f(5.22) This is consistent with a general force law of the form˙p=f=mΩ·v+(˙Ω·S)v(5.23) However,when radiation is ignored,f should presumably depend only on the externalfield F in(4.9).Accordingly,since(4.3)impliesF(z)=F(x)(5.24) for r=0,(5.23)should be replaced by˙p=f=−mm0eF·v+em0S·F(5.25)which is related to(5.22)by˙m=v·f=v·em0S·F=em0S·˙F(5.26)The force law(5.25)appears to be the most general expression for f which holds for arbitrary F and is consistent with the conditions on p.The mean equations of motion are thus determined by consistency conditions without a formal averaging procedure.The ratio m/m0in(5.25)is strange,but a similar ratio appears in the Dirac theory.Of course, it may be that some modification of the expression forΩin(4.9)is needed. Substituting(5.9)into(5.25)and using(5.26)as well as(5.19),we obtainm˙v+v·..−S=emm0F·v(5.27)13。
Nullification of the Nullification
a r X i v :q u a n t -p h /0109034v 1 7 S e p 2001Nullification of the NullificationD.M.Appleby ∗Department of Physics,Queen Mary University of London,Mile End Rd,London E14NS,UKA recent claim by Meyer,Kent and Clifton (MKC),that their models “nullify”the Kochen-Specker theorem,has attracted much comment.In this paper we present a new counter-argument,based on the fact that a classical measurement reveals,not simply a pre-existing value,but pre-existing classical information.In the MKC models measurements do not generally reveal pre-existingclassical information.Consequently,the Kochen-Specker theorem is not nullified.We go on to provea generalized version of the Kochen-Specker theorem,applying to non-ideal quantum measurements.The theorem was inspired by the work of Simon et al and Larsson (SBZL).However,there is aloophole in SBZL’s argument,which means that their result is invalid (operational non-contextualityis not inconsistent with the empirical predictions of quantum mechanics).Our treatment resolvesthis difficulty.We conclude by discussing the question,whether the MKC models can reproduce theempirical predictions of quantum mechanics.I.INTRODUCTION Prior to the work of Meyer [1],the Kochen-Specker theorem [2–6](KS theorem)was generally regarded as one of the key foundational results of quantum mechanics,establishing one of the most important ways in which quantum mechanics enforces a radical departure from the assumptions of classical physics.Classically,a measurement is a process which reveals the pre-existing value of the observable measured.Until recently it was generally accepted that the KS theorem shows that quantum mechanics cannot be interpreted in such a way as to preserve this feature of classical physics.Unfortunately,the KS theorem,as originally formulated,only applies to ideal,or von Neumann measurement processes.In practice,strict ideality is seldom,and perhaps never actually attainable.It follows that the KS theorem,in its original form,is not sufficient to demonstrate a contradiction between classical assumptions and the empirically verifiable predictions of quantum mechanics,regarding the outcome of real,laboratory measurements.At first sight,this may not seem a serious problem.The natural assumption would be that the original KS theorem is a limiting case of a more general theorem,which does apply to non-ideal measurements.We will eventually argue that this is,in fact,the correct assumption.However,the question is greatly complicated by the work of Meyer [1],Kent [7]and Clifton and Kent [8](MKC in the sequel),who have argued that “finite precision measurement nullifies the Kochen-Specker theorem”.MKC’s claimed “nullification”of the KS theorem has given rise to some controversy.It has been discussed by (in chronological order)Cabello [12],Havlicek et al [13],Mermin [14],Appleby [15,16],Simon et al [17],Larsson [18],Simon [19],Cabello [20]and Boyle and Schafir [21].For further discussion of the problem of demonstrating an empiricalcontradiction with the predictions of non-contextual hidden variables theories see Cabello and Garc ´ia-Alcaine [22],Basu et al [23],Cabello [24],Michler et al [25],Simon et al [26]and Cereceda [27].Finally,it should be noted that MKC’s argument was inspired by the previous work of Pitowsky [9],Hales and Straus [10]and Godsil and Zaks [11](we briefly comment on Pitowsky’s work in the conclusion).The arguments in this paper are largely new.They go significantly beyond any that have previously been given.We begin by nullifying MKC’s claimed nullification.That is,we show that MKC’s ingenious mathematical construc-tions,though deeply interesting in their own right,do not invalidate the essential physical point of the KS theorem.Having cleared the conceptual ground,we then go on to establish a generalized version of the KS theorem,which does apply to non-ideal measurements.In order to make the treatment comprehensive we conclude by discussing a number of related questions.In particular,we discuss a recent claim by Cabello [20],that the MKC models do not reproduce the empirically verifiable predictions of quantum mechanics.Our main criticism of MKC’s argument is contained in Sections II–IV.Section II contains some preliminary considerations.MKC assume that an experimenter can never precisely know what has“actually”been measured.This assumption plays an important role in their argument because it enables them to postulate that it is physically impossible to measure each of the observables in a KS-uncolourable set.We show that the assumption depends on some misconceptions regarding non-ideal quantum measurements.Section III contains the crux of the critical part of our argument.The physical point of the KS theorem is to show that quantum measurements cannot generally be interpreted as measurements in the classical sense.A classical measurement is not simply a process which reveals a pre-existing value.Rather,it is a process which reveals a pre-existing piece of classical information,represented by a proposition of the form“observable A took value x”.For this condition to be satisfied it is essential that the value x and observable A both be adequately specified.It appears to MKC that their models nullify the KS theorem because,in their models,a measurement does always reveal the value of something.However,it does not reveal the precise identity of that something.Since the MKC valuations are radically discontinuous this means that measurements do not generally reveal any classical information.Consequently,they cannot generally be regarded as measurements in the classical sense.It follows that the KS theorem is not nullified. Section IV contains some supplementary considerations,which reinforce the conclusion reached in Section III. The argument just outlined goes beyond the arguments given in our previous papers because it shows that MKC’s claim,to have nullified the KS theorem,is simply false.In Appleby[15,16]we showed that the MKC models are highly non-classical.In fact,we showed in ref.[16]that the MKC models exhibit a novel kind of contextuality,which is even more strikingly at variance with classical assumptions than the usual kind of contextuality,featuring in the KS theorem.Consequently,the argument in ref.[16]is,by itself,sufficient to refute MKC’s suggestion,that their models provide a classical explanation of non-relativistic quantum mechanics.The argument in ref.[15],though less clear-cut,lends additional support to this conclusion.However,neither argument directly bears on MKC’s claim, to have nullified the KS theorem as such.Indeed,at the time we wrote these papers it appeared to us that MKC actually had“nullified the KS theorem strictly so-called”(as we put it).It is important to establish thatfinite precision does unequivocally not nullify the KS theorem because,in the absence of such a demonstration,it is impossible to achieve an unobstructed understanding of contextuality,as it applies in a real,laboratory setting.The question has immediate,practical relevance,in view of the current interest in experimental investigations of contexuality[17–19,22–27].Having cleared the conceptual ground in Sections II–IV we go on,in Section V,to formulate and prove a generalized KS theorem,applying to non-ideal measurements.The argument in this section is inspired by the argument of Simon et al[17,19]and Larsson[18](SBZL in the sequel).However,there are some important differences.SBZL work with an “operational”concept of contextuality.Their motive for introducing this concept is to circumvent MKC’s argument. Clearly,the motive no longer applies,once it is established that MKC’s argument is invalid.We are consequently able to formulate a generalized KS theorem in terms of the ordinary concept of contextuality.This has several advantages. In thefirst place it means that the result we prove is a straightforward generalization of the ordinary KS theorem. In the second place,there is a loophole in SBZL’s argument,which means that their result is actually incorrect.As we show in Appendix C,by means of a counter-example,operational non-contextuality is not inconsistent with the empirical predictions of quantum mechanics.The modified argument we give in Section V resolves this difficulty. Lastly,the result we prove implies a significantly different conclusion,as to the conditions which must be satisfied in order refute non-contextual theories experimentally.Sections II–V contain the main part of our argument.In Sections VI–VIII we address a number of related questions. In Section VI we briefly discuss the POVM arguments given by Kent[7]and Clifton and Kent[8].In Section VII we give an improved version of the argument in Appleby[15].In thefirst place we have improved the argument so as to take account of the points made in Section II of this paper.In the second place,the version we give now does not involve joint measurements of non-commuting observables.In the third place,we have strengthened the argument,using ideas derived from the subsequent work of SBZL.In Section VIII we discuss the question,whether there actually exists a complete theory of MKC type which is empirically equivalent to quantum mechanics.In particular,we discuss a recent claim by Cabello[20],that theories of this type make experimentally testable predictions which conflict with those of quantum mechanics.We argue that, although Cabello makes some very pertinent points,the question remains open.II.WHAT IS“ACTUALLY”MEASURED?MKC’s argument partly depends on a misconception,regarding non-ideal quantum measurements.We begin our critical discussion by clarifiying this point.The criticisms which follow are preliminary to our main critical argument, contained in the next section.MKC postulate that there are many observables which it is physically impossible to measure.If an experimenter attempts to measure an observable ˆA in this forbidden set,then what is actually measured is a slightly different observable ˆB.The measurement reveals the pre-existing value of the observable ˆB which is actually measured.It can be seen from this that MKC tacitly assume:Property 1For a given finite precision measuring apparatus there is a single,uniquely defined observable,which is the only observable that is “actually”measured.Property 2The nominal observable,which is recorded in the experimenter’s notebook as having beenmeasured,is typically not the observable which is “actually”measured.We will show that these assumptions are unjustified.Before proceeding further,let us clarify the meaning of theterm “nominal observable”,as it appears in the above statement.The term essentially corresponds to Simon et al ’s [17]“switch position”.However,unlike Simon et al we are not appealing to the concept of an “operational observable”(i.e.the concept that different instruments define different observables—see Appendix C).We are simply recognizing the fact that the complete record of a measurement (quantum or classical)must include,not only a specification of the value obtained,but also a specification of the observable measured.Suppose,for instance,that an experimenter makes 1,000different measurements of 1,000different observables.If the experimenter simply writes down a list of 1,000numbers,without any indication as to how those numbers were obtained,then the record will obviously be useless (this point,that the outcome of a measurement [quantum or classical]is,not simply a value ,but a determinate proposition ,will play a central role in the argument of the next section).It should be stressed that,although we refer,for convenience,to the actions of a human experimenter,the same point would apply if the measurements were performed by a completely arbitrary “information gathering and utilizing system”[28].If the system is to utilize the information it acquires then it must record,in its memory area,a specification of the observables measured,as well as the values obtained.The nominal observable need not be specified precisely.Rather than recording the information “observable ˆS z was measured and value +1/2obtained”,one might instead record the information “observable n ·ˆSwas measured,for n ∈U ,and value +1/2obtained”(where U is some subset of the unit 2-sphere).In the following we will assume that the nominal observable is specified precisely,because this is legitimate,and because it happens to be the usual practice.However,the argument could easily be modified so as to allow for the possibility that the nominal observable is only specified partially.Let us now consider MKC’s assumption that,for each quantum measurement,there is one,and only one observable which is “actually”measured.The fact that this assumption is unjustified becomes apparent when one takes into view (as MKC do not)the detailed physical implementation of an abstract quantum measurement scheme.zFIG.1.Schematic illustration of the Stern-Gerlach arrangement.The surface irregularities of the pole pieces,and asym-metries in the environment,mean that the field is not perfectly symmetric.There is no single direction which represents the alignment of the field more truly than any other.It follows that there is no single spin component which is measured more truly than any other.Consider,for example,a non-ideal measurement of the spin component n ·ˆSusing a Stern-Gerlach apparatus.The vector n is determined by the axis of symmetry of the arrangement.MKC take it that,in such a case,although the axisis not precisely known to the experimenter,there does actually exist a single,sharply defined axis of symmetry;and that there correspondingly exists a single,sharply defined spin component n·ˆS which is the only spin component thatis actually measured.However,this is to overlook the fact that the magneticfield will not,in practice,be perfectly symmetric(due to irregularities in the shape of the pole-pieces,asymmetries in the placement of surrounding objects,etc.).As illustrated in Fig.1the numerous slight departures from perfect symmetry introduce some unavoidable “fuzziness”into the concept“axis of symmetry of thefield”.Under these conditions there is no single vector n whichrepresents the axis of symmetry more truly than any other.This being so,there are no obvious grounds for picking out any single spin component as the only component which is“actually”measured.A measurement[30]of n·ˆS is a process which discriminates the eigenstates of n·ˆS.Given an unknown eigenstate of n·ˆS,an ideal measuring instrument will inform the experimenter,with certainty,which particular eigenstate it was.A non-ideal measuring instrument will inform the experimenter,with high probability,which particular eigenstate it was.This is the definition of a quantum measurement process[30]:if a process performs the function,of discriminating the eigenstates of an operator,then it is a measurement of that operator,ideal or non-ideal as the case may be.Of course,it is seldom,if ever the case that a real laboratory instrument is strictly ideal,and the Stern-Gerlach arrangement is no exception to this rule.It is important,furthermore,to note that,not only does a Stern-Gerlacharrangement fail to perform an ideal measurement of the nominal spin component,which is recorded in the experi-menter’s notebook as having been measured;it fails to perform an ideal measurement of any other spin componenteither(for a discussion of the unavoidable sources of non-ideality in Stern-Gerlach measurements see Busch et al[31]). The arrangement may,however,be used to perform non-ideal measurements(which is the most that can reasonably be demanded of a real,laboratory instrument).We can characterize the degree of non-ideality in quantitative terms by considering the probability that the process will fail to correctly identify a given eigenstate.Suppose that the arrangement is used to measure the componentn·ˆS for a spin-1/2particle.Let p+−(n)(respectively p−+(n))be the probability that,when the system particle is initially in the spin up(respectively spin-down)eigenstate of n·ˆS the measurement outcome is−1/2(respectively +1/2).Then it may be said that,the smaller these probabilities are,the more nearly ideal[32]the measurement.There is no vector n for which p+−(n)=p−+(n)=0(see Busch et al[31]).There are,however,vectors n for which the failure probabilities p+−(n),p−+(n)are both small.If n satisfies this condition the arrangement can be used to discriminate the eigenstates of n·ˆS with a high degree of reliability.In other words,it can be used to performnon-ideal measurements of n·ˆS.The failure probabilities p+−(n),p−+(n)vary continuously with n.This means that,if the probabilities are smallfor one vector n,then they will also be small for every other vector which is close to n.It follows that,for a given arrangement,there are infinitely many different spin components which are all non-ideally measured.Furthermore theclass of observables which are non-ideally measured includes the nominal observable(provided it is correctly recorded, and provided the instrument does not malfunction).We began our discussion of the Stern-Gerlach arrangement by noting that,in practice,the arrangement will not havea sharply defined axis of symmetry.It can now be seen that,even if the symmetry were exact,MKC’s assumption, that there is only one observable that is“actually”measured,would still not be justified.It is true that there might(perhaps)then be a well-defined,natural sense in which the measurement of n·ˆS was“most nearly ideal”when n was parallel to the axis of symmetry.But it would not follow that a neighbouring component n′·ˆS,defined by a vector n′which was not exactly parallel to the axis of symmetry,was not“actually”measured at all.A process does not have to be precisely optimal in order to count as a measurement.If the probabilities p+−(n) and p−+(n)become slightly larger,then that simply means that the process does not discriminate eigenstates quite so efficiently as before.It does not mean that the process thereby ceases to be a measurement.The real photon detectors which have been constructed to date are imperfect,in that there is a non-zero probabilitythat the detector will fail to register the presence of a photon.This does not mean that a real photon detector does not“actually”detect photons at all.Furthermore,a photon detector whose performance becomes slightly degraded, so that it is no longer quite so efficient as before,does not,on that account,cease to be a photon detector.In general,however,there is no unambiguous,non-arbitrary,physically well-motivated way to define the concept“unique spin component for which the measurement is most nearly ideal”.For instance,minimizing the function (1/2) p+−(n)+p−+(n) ,and minimizing the function (1/2) p+−(n)2+p−+(n)2 1/2leads to definitions of this concept which are,in general,incompatible(quite apart from the fact that the minima may not be unique).In the general case there are many spin components which all have an equally valid claim to the status“spin component for which the measurement is most nearly ideal”.Thus far we have been considering the case of the Stern-Gerlach apparatus.However,it is shown in Appendix A that the above criticisms apply much more generally,to the case of any approximate von Neumann measurement (i.e.any non-ideal measurement described by a unitary operator which is close to a unitary operator describing a von Neumann measurement).This provides strong grounds for the assertion that Properties1and2assumed by MKCshould be replaced byProperty1′For a givenfinite precision measuring apparatus there are infinitely many different observables which are non-ideally measured.Property2′The nominal observable is among the observables which are non-ideally measured(provided the nominal observable is recorded correctly,and provided there is no malfunction in themeasuring apparatus).It may be worth remarking that these statements are also valid classically.Consider,for example a classical measurement of the velocity component v z of some macroscopic object,using the Doppler shift of the radiation it emits.In order to perform this measurement it is necessary that the line running from source to detector should be parallel to the z-axis.The fact that source and detector are both extended objects,which are not perfectly symmetric, means that,in practice,this line is not even precisely defined,let alone precisely known.However,this does not imply that v z may not be the component which is“actually”measured.Rather,it is to be regarded as one of the sources of error in the measurement of v z.A classical measurement of a classical observable A is a process which reveals, with a high degree of reliability,and to a high degree of accuracy,the pre-existing value of A.On this definition the procedure described effects a non-ideal classical measurement of n·v for every n which is nearly(but perhaps not precisely)parallel to the z-axis.The effect of the argument in this section is to restore the natural assumption of most physicists,that the observable which an experimenter records as having been non-ideally measured is also measured in fact(provided the experimenter does not make a mistake,and provided there is no malfunction in the measuring apparatus).Let us now consider the bearing which this has on MKC’s argument.Our criticisms do not invalidate MKC’s postulate,that a measurement ofˆA may reveal the value of some other,neighbouring observableˆB.They do, however,show that MKC cannot justifiably claim thatˆB is the only observable which is“actually”measured,and thatˆA is not really measured at all.MKC cannot even claim thatˆB is the uniquely defined observable for which the measurement is most nearly ideal.In general,the observableˆB has no preferred status.Properly understood,MKC’s postulate comes to this:that the hidden dynamics somehow selects an essentially arbitrary observableˆB≈ˆA,whose value it then reveals.The point is important because,once MKC’s proposal is formulated correctly,it becomes easier to see the fallacy in their argument.III.NULLIFICATION OF THE NULLIFICATIONThis section contains the main part of our critical argument.The KS theorem showsProposition1No hidden variables interpretation can have the property that,for every observableˆA,an ideal measurement ofˆA always reveals the pre-existing value ofˆA.MKC’s claim,when appropriately re-formulated,so as to remove any unjustified appeal to the concept of“the unique observable which is actually measured”,is that this proposition is“nullified”byProposition2There do exist hidden variables interpretations with the property that,for every observable ˆA,afinite precision measurement ofˆA always reveals the pre-existing value of some unknownobservableˆB in a small neighbourhood ofˆA.In considering this claim thefirst problem we face is that the term“nullified”is not part of the standard lexicon of science and mathematics,and so its meaning is potentially ambiguous.Propositions1and2are logically consistent, so there can be no question of the KS theorem actually being refuted.It is not immediately apparent what,exactly, is meant by the claim that Proposition2“nullifies”Proposition1,without contradicting it.In this connection it should be noted that the reference tofinite precision measurements,on which MKC themselves place much emphasis,is actually irrelevant.MKC postulate that,for each non-ideal measuring apparatus,the hidden dynamics somehow arbitrarily selects an observableˆB,close to the nominal observable,whose value the measurement then reveals.One could,with equal justification,postulate that the same is true for each ideal measuring apparatus: implyingProposition2′There do exist hidden variables interpretations with the property that,for every observableˆA, an ideal measurement ofˆA always reveals the pre-existing value of some unknown observableˆB in a small neighbourhood ofˆA.If the KS theorem is nullified by Proposition2,then it must,presumably,also be nullifed by Proposition2′.It would seem to follow that such nullification as there might be cannot be attributed specifically to thefinite precision.In the following we will take the claim to be that the MKC models invalidate what had previously been regarded as the key physical implication of the KS theorem:namely,the implication that quantum measurements cannotsystematically be interpreted as measurements in the classical sense.In the MKC models a measurement does not always reveal the value of the nominal observable.It does,however,always reveal the value of an observablewhich is extremely close to the nominal observable.MKC assume that is enough to recover the classical concept of measurement.The question we now have to decide is whether MKC are correct to assume that a process which does not reveal the pre-existing value of the nominal observable may still count as a measurement in the classical sense.We will arguethat they are not.The crucial point to realize is that a classical measurement is not simply a process which reveals a pre-existingvalue.Rather,it is a process which reveals a pre-existing piece of classical information,represented by a proposition of the form“observable A took value x”.For this to be true it is essential that the observable A and the value x bothbe adequately specified.A process which reveals the value of some completely unknown observable does not reveal any classical information,and is not a classical measurement.The following example may serve to illustrate this point.Suppose that an experimenter is given a sealed box, containing100individually labelled rods.Suppose that,for each integer n=1,...,100the box contains exactly onerod of length n cm.Suppose that the experimenter,without looking inside the box,uses a random number generator to select a particular integer1≤n0≤100.It could be said that this procedure reveals the length of one of the rods in the box.However,it does not reveal the identity of that rod.Consequently,the process does not reveal any classicalinformation,and so it cannot be considered a classical measurement.It may,atfirst sight,seem that these remarks do not apply to the situation envisaged by MKC since,in theirmodels,the observableˆB,whose value is revealed by the measurement,is not completely unknown.In fact,it is assumed thatˆB is extremely close to the nominal observableˆA:so there is a sense in whichˆB is specified very precisely.However,it is not specified precisely enough.The point here is that the MKC valuations are radically discontinuous.If the valuation could be assumed continuousin the vicinity ofˆA,then a process which revealed the value of some neighbouring observableˆB could be regarded as a measurement in the classical sense.However,there is in fact always a region on which the valuation is highly discontinuous,and so this assumption would not generally be justified.Consider,for example,the colourings of the unit2-sphere S2discussed by Meyer[1](and described in more detailby Havlicek et al[13]).These colourings have the property that,given any n∈S2,every neighbourhood of n contains infinitely many points evaluating to0,and infinitely many points evaluating to1.Suppose,now,that one performs a finite precision measurement of the nominal observable(n·ˆS)2and obtains(say)the value1.Then one knows that, there is a point n′in some small neighbourhood U of n which evaluates to1.However,U contains infinitely many points evaluating to1,and infinitely many points evaluating to0.The process does not reveal any more information, as to which particular point takes which particular value,than could be obtained by tossing a coin.Consequently, it cannot be considered a classical measurement.It is not a classical measurement for the same reason that random number generators cannot be used to make classical measurements of length.Essentially the same criticism applies to an arbitrary model of MKC type.In the general case the highly discon-tinuous behaviour just discussed may not occur everywhere,on the whole of S2.However,there will always be a non-empty open subset of S2on which such behaviour occurs.The detailed proof of this statement is given in Appendix B.We will confine ourselves here to summarising the main points.Consider an arbitrary MKC colouring f:S20→{0,1}.Here S20is any dense,KS-colourable subset of the unit 2-sphere S2having the property that the set of triads contained in S20is dense in the space of all triads(MKC assume that S20is also countable;however the argument which follows does not depend on this assumption).We define the discontinuity region D⊆S2to consist of those points n∈S2with the property that each neigh-bourhood of n contains infinitely many points evaluating to0,and infinitely many points evaluating to1(it is not assumed that n itself∈S20).We define the continuity region C⊆S2to consist of those points n∈S2with the property that f is continuous on U∩S20,for some neighbourhood U of n.In the case of the Meyer colourings C=∅and D=S2.However,this is not true generally.It is readily verified that these regions partition S2(the complete unit2-sphere,not just S20)into two disjoint subsets:C∪D=S2and C∩D=∅.It is also readily verified that C is open and D is closed.The key result,proved in Appendix B,is that C is not only open,but also KS-colourable.Furthermore,the KS-colouring f defined on C∩S20uniquely extends to a continuous,induced KS-colouring¯f:C→{0,1}.We show in Appendix B that this result places significant constraints on the minimum size of the discontinuity region.In thefirst place,D must have non-empty interior(i.e.it must contain a non-empty open subset—implying that。
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constraint
sions which do not involve default negated literals, or are incomplete 15]. The novelty of our procedure is the simultaneous treatment of positive and negative recursions, together with general constraint handling features used to implement constructive negation, enabling us to address in this way the oundering problem in its full generality. For lack of space we only provide a high-level description of our procedure focusing on its major features.
2
1 Introduction
The applications of logic programming are wide-ranging, namely for knowledge representation. Nevertheless, a mechanism for explicitly declaring the falsity of propositions is not available in the restricted language of normal logic programs. The importance of extending logic programming with an explicit form of non-classical negation, \:", alongside the usual default or implicit one, has been stressed for use in deductive databases, knowledge representation, and non-monotonic reasoning. This has been pointed out by several authors, and an enhanced language and semantics have been proposed. For a recent study of this explicit form of negation consult 2]. However, the introduction of explicit negation permits and requires being able to reason with, or at least detect, contradictory information. Indeed, information is not only normally incomplete but contradictory as well. The idea of introducing paraconsistent reasoning in logic programming is fairly recent. Constraint logic programming adds to knowledge espressivity by merging two declarative programming paradigms: constraint solving and logic programming. A constraint logic program is a logic program parametrized with a constraint system de ning the meaning of several primitive predicates (the basic constraints). The advantage of such frameworks is the smooth integration of a domain associated with constraints, often an arithmetic domain, into a logic programming language with clear semantical properties, namely the equivalence of operational, model theoretic, and xpoint semantics. The starting point of this paper is our paraconsistent well-founded semantics with explicit negation (WFSX p ) 1, 8], to which we add general constraints. In order to have an usable working system several issues must be addressed: soundness, completeness, termination and e ciency. We achieve this by relying on a proof procedure which integrates tabulation and constraint handling features and, in particular, constructive negation. The tabulation mechanisms provide a solution to the problems of termination of the proof procedure, namely negative and positive recursions. The integration of constraint technology in tabulation procedures is not new, see for instance 6]. However, the systems present in the literature only take care of positive recursions, i.e. recur-
bessiere@lirmm.fr2 NICTAandUNSW,Sydney,Australia
Among,Common and Disjoint Constraints Christian Bessiere1,Emmanuel Hebrard2,Brahim Hnich3,Zeynep Kiziltan4,and Toby Walsh21LIRMM,CNRS/University of Montpellier,Francebessiere@lirmm.fr2NICTA and UNSW,Sydney,Australia{ehebrard,tw}@.au3Izmir University of Economics,Turkeybrahim.hnich@.tr4University of Bologna,Italyzkiziltan@deis.unibo.itAbstract.Among,Common and Disjoint are global constraints use-ful in modelling problems involving resources.We study a number of vari-ations of these constraints over integer and set variables.We show howcomputational complexity can be used to determine whether achievingthe highest level of consistency is tractable.For tractable constraints,wepresent a polynomial propagation algorithm and compare it to logical de-compositions with respect to the amount of constraint propagation.Forintractable cases,we show in many cases that a propagation algorithmcan be adapted from a propagation algorithm of a similar tractable one.1IntroductionGlobal constraints are an essential aspect of constraint programming.See,for example,[8,3,9,2].They specify patterns that occur in many problems,and ex-ploit efficient and effective propagation algorithms to prune search.In problems involving resources,we often need to constrain the number of variables taking particular values.For instance,we might want to limit the number of night shifts assigned to a given worker,to ensure some workers are common between two shifts,or to prevent any overlap in shifts between workers who dislike each other. The Among,Common and Disjoint constraints respectively are useful in such circumstances.The Among,Common and Disjoint constraints are useful in such circumstances.The Among constraint wasfirst introduced in CHIP to model resource allo-cation problems like car sequencing[3].It counts the number of variables using values from a given set.A generalization of the Among and AllDifferent con-straints is the Common constraint[2].Given two sets of variables,this counts the number in each set which use values from the other set.A special case of the Common constraint also introduced in[2]is the Disjoint constraint.This ensures that no value is common between two sets of variables.We study these three global constraints as well as seven other variations over integer and set vari-ables.For each case,we present a polynomial propagation algorithm,and identify when achieving a higher level of local consistency is intractable.For example,B.Hnich et al.(Eds.):CSCLP2005,LNAI3978,pp.29–43,2006.c Springer-Verlag Berlin Heidelberg200630 C.Bessiere et al.rather surprisingly,even though the Disjoint constraint is closely related to (but somewhat weaker than)the AllDifferent constraint,it is NP-hard to achieve generalised arc consistency on it.The rest of the paper is oragnised as follows.Wefirst present the necessary formal background in Section2.Then,in Section3and Section4we study vari-ous generalisations and specialisations of the Among,Common,and Disjoint constraints on integer and set variables.Finally,we review related work in Section5before we conclude and present our future plans in Section6.2Formal BackgroundA constraint satisfaction problem consists of a set of variables,each with afinite domain of values,and a set of constraints specifying allowed combinations of values for given subsets of variables.A solution is an assignment of values to the variables satisfying the constraints.We consider both integer and set variables.A set variable S can be represented by a lower bound lb(S)which contains the definite elements and an upper bound ub(S)which contains the definite and po-tential elements.We use the following notations:X,Y,N,and M(possibly with subscripts)denote integer variables;S and T(again possibly with subscripts) denote set variables;S(possibly with a subscript)and K denote sets of inte-gers;and v and k(possibly with a subscript)denote integer values.We write D(X)for the domain of a variable X.For integer domains,we write min(X) and max(X)for the minimum and maximum elements in D(X).Throughout the paper,we consider constraint satisfaction problems in which a constraint contains no repeated variables.Constraint solvers often search in a space of partial assignments enforcing a local consistency property.A bound support for a constraint C is a partial assignment which satisfies C and assigns to each integer variable in C a value between its minimum and maximum,and to each set variable in C a set between its lower and upper bounds.A bound support in which each integer variable takes a value in its domain is a hybrid support.If C involves only integer variables, a hybrid support is a support.A constraint C is bound consistent(BC)ifffor each integer variable X,min(X)and max(X)belong to a bound support,and for each set variable S,the values in ub(S)belong to S in at least one bound support and the values in lb(S)are those from ub(S)that belong to S in all bound supports.A constraint C is hybrid consistent(HC)ifffor each integer variable X,every value in D(X)belongs to a hybrid support,and for each set variable S, the values in ub(S)belong to S in at least one hybrid support and the values in lb(S)are those from ub(S)that belong to S in all hybrid supports.A constraint C over integer variables is generalized arc consistent(GAC)ifffor each variable X,every value in D(X)belongs to a support.If all variables in C are integer variables,HC is equivalent to GAC,whilst if all variables in C are set variables, HC is equivalent to BC.Finally,we will compare local consistency properties applied to(sets of)logically equivalent constraints.A local consistency property Φon C1is strictly stronger thanΨon C2iff,given any domains,Φremoves all valuesΨremoves,and sometimes more.Among,Common and Disjoint Constraints31 3Integer Variables3.1Among ConstraintThe Among constraint counts the number of variables using values from a given set[3].More formally,we have:Among([X1,..,X n],[k1,..,k m],N)iffN=|{i|∃j.X i=k j}| For instance,we can use this constraint to limit the number of tasks(variables) assigned to a particular resource(value).Enforcing GAC on such a constraint is polynomial.Before we give an algorithm to do this,we establish the following theoretical results.Lemma1.Given K={k1,..,k m},lb=|{i|D(X i)⊆K}|,and ub=n−|{i| D(X i)∩K=∅}|,a value v∈D(N)is GAC for Among ifflb≤v≤ub. Proof.At most ub variables in[X1,..,X n]can take a value from K and lb of these take values only from K.Hence v is inconsistent if v<lb or v>ub.We now need to show any value between lb and ub is consistent.We have ub−lb variables that can take a value from K as well from outside K.A support for lb≤v≤ub can be constructed by assigning v variables to a value from K and ub−v variables to a value from outside K.⊓⊔Lemma2.Given K={k1,..,k m},lb=|{i|D(X i)⊆K}|,ub=n−|{i| D(X i)∩K=∅}|,and lb≤min(N)≤max(N)≤ub,a value in D(X i)may not be GAC for Among ifflb=min(N)=max(N)or min(N)=max(N)=ub.Proof.The variables[X1,..,X n]can be divided into three categories:1)those whose domain contains values only from K(lb of them),2)those whose domain contains both values from K and from outside(ub−lb of them),and3)those whose domain does not intersect with K(n−ub of them).If lb=min(N)= max(N)then exactly lb variables must take a value from K.These variables can then only be those of thefirst category and thus K cannot be in the domains of the second category.If min(N)=max(N)=ub then exactly ub variables must take a value from K.These variables can then only be those of thefirst and the second category and thus any value v∈K cannot be in the domains of the second category.We now need to show this is the only possibility for inconsistency.Consider an assignment to the constraint.Due to the variables of thefirst and the third category we have lb values from K and n−ub values from outside K.If lb<max(N)then in the second category we can have at least one variable assigned to a value from K,the rest assigned to a value outside K and satisfy the constraint.Similarly,if min(N)<ub then in the second category we can have at least one variable assigned to a value outside of K,the rest assigned to a value from K and satisfy the constraint.Hence,all values are consistent when lb<max(N)or min(N)<ub.⊓⊔We now give an algorithm for the Among constraint.32 C.Bessiere et al.lb:=|{i|D(X i)⊆K}|;1ub:=n−|{i|D(X i)∩K=∅}|;2min(N):=max(min(N),lb);3max(N):=min(max(N),ub);4if(max(N)<min(N))then fail;5if(lb=min(N)=max(N))then6foreach X i.D(X i)⊆K do D(X i):=D(X i)\K;if(min(N)=max(N)=ub)then7foreach X i.D(X i)∩K=∅do D(X i):=D(X i)∩K;Among,Common and Disjoint Constraints33 Even if GAC on Among can be maintained by a simple decomposition,the presented algorithm is useful when we consider a number of extensions of the Among constraint.An interesting extension is when we count not the variables taking some given values but those taking values taken by other variables.This is useful when,for example,the resources to be used are not initially known.We consider here two such extensions in which we replace[k1,..,k m]either by a set variable S or by a sequence of variables[Y1,..,Y m]Among([X1,..,X n],S,N)holds iffN variables in X i take values in the set S.That is,N=|{i|X i∈S}|.Enforcing HC on this constraint is NP-hard in general.Theorem2.Enforcing HC on Among([X1,..,X n],S,N)is NP-hard. Proof.We reduce3-Sat to the problem of deciding if such an Among con-straint has a satisfying assignment.Finding hybrid support is therefore NP-hard.Consider a formulaϕwith n variables(labelled from1to n)and m clauses.Let k be m+n+1.To construct the Among constraint,we create 2k+1variables for each literal i in the formula such that X i1..X ik∈{i}, X i(k+1)..X i(2k)∈{−i},and X i(2k+1)∈{i,−i}.We create a variable Y j for each clause j inϕand let Y j∈{x,−y,z}where the j th clause inϕis x∨¬y∨z. We let N=n(k+1)+m and{}⊆S⊆{1,−1,..,n,−n}.The constraint Among([X11,..,X1(2k+1),..,X n1,..,X n(2k+1),Y1,..,Y m],S,N)has a solution iffϕhas a satisfying assignment.♥In Algorithm2,we give a propagation algorithm for this Among constraint. Notice that we assume all values are strictly positive.We highlight the differences with Algorithm1.Thefirst modification is to replace each occurrence of K by either lb(S)or ub(S).As a consequence,instead of a single lower bound and upper bound on N,we have now two pairs of bounds,one under the hypothesis that S isfixed to its lower bound(lb[0]and glb[0]),and one under the hypothesis that S isfixed to its upper bound(lub[0]and ub[0]).Moreover,in loop1,we compute the contingent values of lb(resp.ub)when a value v is added to lb(S) (resp.removed from ub(S))and store the results in lb[v](resp.ub[v]).These arrays are necessary for pruning N(lines3,4,6,7),when the minimum(resp. maximum)value of N cannot be achieved with the current lower(resp.upper) bound of S(conditionals2and5).In this case,we know that at least one of these values must be added to lb(S)(resp.removed from ub(S)).Therefore the smallest value lb[v](resp.greatest value ub[v])is a valid lower bound(resp.upper bound)on N.We also use them for pruning S(lines8and9).Finally,we need to compute lb and ub,as they may have been affected by the pruning on S.This is done in line10.The worst case time complexity is unchanged,as loop1can be done in O(nd).The level of consistency achieved by this propagation algorithm is incompa-rable to BC.The following example shows that BC is not stronger:X1∈{2,3}, X2∈{2,3},X3∈{1,2,3,4},lb(S)=ub(S)={2,3},min(N)=max(N)=2. The algorithm will prune{2,3}from X3,whereas a BC algorithm will not do34 C.Bessiere et al.lb[0]:=|{X i|D(X i)⊆lb(S)}|;glb[0]:=n−|{X i|D(X i)∩lb(S)=∅}|;ub[0]:=n−|{X i|D(X i)∩ub(S)=∅}|;lub[0]:=|{X i|D(X i)⊆ub(S)}|;foreach v∈ub(S)\lb(s)do1lb[v]:=|{X i|D(X i)⊆(lb(S)∪{v})}|;ub[v]:=n−|{X i|D(X i)∩(ub(S)\{v})=∅}|;if glb[0]<min(N)then2LB:={lb[v]|v∈(ub(S)\lb(S))};3if(LB=∅)then min(N)=min(LB);4elsemin(N):=max(min(N),lb[0]);if lub[0]>max(N)then5UB:={ub[v]|v∈(ub(S)\lb(S))};6if(UB=∅)then max(N)=max(UB);7elsemax(N):=min(max(N),ub[0]);if(max(N)<min(N))then fail;lb(S):=lb(S)∪{v|ub[v]<min(N)};8ub(S):=ub(S)\{v|lb[v]>max(N)};9if(min(N)=max(N))then10lb:=|{i|D(X I)⊆lb(S)}|;ub:=|{i|D(X I)∩ub(S)=∅}|;if(lb=min(N))thenforeach X i.D(X i)⊆lb(S)do D(X i):=D(X i)\lb(S);if(ub=max(N))thenforeach X i.D(X i)∩ub(S)=∅do D(X i):=D(X i)∩ub(S);Among,Common and Disjoint Constraints35 The level of consistency achieved by this decomposition is also incomparable to BC.The example which demonstrates the incomparability of Algorithm2and BC also shows that the decomposition is incomparable to BC.It remains an open question,however,whether BC on such a constraint is tractable or not.We now consider the second generalization.Among([X1,..,X n],[Y1,..,Y m],N) holds iffN variables in X i take values in common with Y j.That is,N= |{i|∃j.X i=Y j}|.As before,we cannot expect to enforce GAC on this con-straint.Theorem3.Enforcing GAC on Among([X1,..,X n],[Y1,..,Y m],N)is NP-hard. Proof.We again use a transformation from3-Sat.Consider a formulaϕwith n variables(labelled from1to n)and m clauses.We construct the constraint Among([Y1,..,Y m],[X1,..,X n],M)in which X i represents the variable i and Y j represents the clause j inϕ.We let M=m,X i∈{i,−i}and Y j∈{x,−y,z} where the j th clause inϕis x∨¬y∨z.The constructed Among constraint has a solution iffϕhas a model.♥To propagate Among([X1,..,X n],[Y1,..,Y m],N),we can use the following de-composition:Among([X1,..,X n],[Y1,..,Y m],N)iffAmong([X1,..,X n],S,N)∧{Y j}=Sj∈{1,..,m}We can therefore use the propagation algorithm proposed for Among([X1, ..,X n],S,N).However,even if we were able to enforce HC on the decomposition (which is NP-hard in general to do),we may not make the original constraint GAC.Theorem4.GAC on Among([X1,..,X n],[Y1,..,Y m],N)is strictly stronger than HC on the decomposition.Proof:It is at least as strong.To show the strictness,consider Y1∈{1,2,3}, X1∈{1,2},X2∈{1,2,3},N=2.We have{}⊆S⊆{1,2,3},hence the decom-position is HC.However,enforcing GAC on Among([X1,X2],[Y1],N)prunes3 from Y1and X2.♥Again,we still do not know whether BC on such a constraint is tractable or not.3.2Common ConstraintA generalization of the Among and AllDifferent constraints introduced in[2]is the following Common constraint:Common(N,M,[X1,..,X n],[Y1,..,Y m])iffN=|{i|∃j.X i=Y j}|∧M=|{j|∃i.X i=Y j}|36 C.Bessiere et al.That is,N variables in X i take values in common with Y j and M variables in Y j take values in common with X i.Hence,the AllDifferent constraint is a special case of the Common constraint in which the Y j enumerate all the values j in X i,Y j={j}and M=n.Not surprisingly,enforcing GAC on Common is NP-hard in general,as the result immediately follows from the intractability of the related Among constraint.Theorem5.Enforcing GAC on Common is NP-hard.Proof.Consider the reduction in the proof of Theorem3.We let N∈{1,..,n}. The constructed Common constraint has a solution iffthe original3-Sat prob-lem has a model.♥As we have a means of propagation for Among([X1,..,X n],[Y1,..,Y m],N),we can use it to propagate the Common constraint using the following decomposi-tion:Common(N,M,[X1,..,X n],[Y1,..,Y m])iffAmong([X1,..,X n],[Y1,..,Y m],N)∧Among([Y1,..,Y m],[X1,..,X n],M)In the next theorem,we prove that we might not achieve GAC on Common even if we do so on Among.Theorem6.GAC on Common is strictly stronger than GAC on the decompo-sition.Proof:It is at least as strong.To show the strictness,consider N=2,M=1, X1,Y1∈{1,2},X2∈{1,3},Y2∈{1},and Y3∈{2,3}.The decomposition is GAC.However,enforcing GAC on Common(N,M,[X1,X2],[Y1,Y2,Y3])prunes 2from X1,3from X2,and1from Y1.♥Similar to the previous cases,the tractability of BC on such a constraint needs further investigation.3.3Disjoint ConstraintWe may require that two sequences of variables be disjoint(i.e.have no value in common).For instance,we might want the sequence of shifts assigned to one person to be disjoint from those assigned to someone who dislikes them.The Disjoint([X1,..,X n],[Y1,..,Y m])constraint introduced in[2]is a special case of the Common constraint where N=M=0.It ensures X i=Y j for any i and j. Surprisingly,enforcing GAC remains intractable even in this special case. Theorem7.Enforcing GAC on Disjoint is NP-hard.Proof:We again use a transformation from3-Sat.Consider a formulaϕwith n variables(labelled from1to n)and m clauses.We construct the Disjoint con-straint in which X i represents the variable i and Y j represents the clause j inϕ.Among,Common and Disjoint Constraints37 We let X i∈{i,−i}and Y j∈{−x,y,−z}where the j th clause inϕis x∨¬y∨z. The constructed Disjoint constraint has a solution iffϕhas a model.♥An obvious decomposition of the Disjoint constraint is to post an inequality constraint between every pair of X i and Y j,for all i∈{1,..,n}and for all j∈{1,..,m}.Not surprisingly,the decomposition hinders propagation(other-wise we would have a polynomial algorithm for a NP-hard problem).Theorem8.GAC on Disjoint is strictly stronger than AC on the binary de-composition.Proof:It is at least as strong.To show the strictness,consider X1,Y1∈{1,2}, X2,Y2∈{1,3},Y3∈{2,3}.Then all the inequality constraints are AC.However, enforcing GAC on Disjoint([X1,X2],[Y1,Y2,Y3])prunes2from X1,3from X2, and1from both Y1and Y2.♥This decomposition is useful if we want to maintain BC on Disjoint. Theorem9.BC on Disjoint is equivalent to BC on the decomposition.Proof.It is at least as strong.To show the equivalence,we concentrate on X i’s, but the same reasoning applies to Y j’s.Given X k where k∈{1,..,n},we show that for any bound b k of X k(b k=min(X k)or b k=max(X k))there exists a bound support containing it.We partition the integers as follows.S X contains all integers v such that∃X i,D(X i)={v},S Y contains all integers w such that ∃Y j,D(Y j)={w},and T contains the remaining integers.T inherits the total ordering on the integers.So,we can partition T in two sets T1and T2such that no pair of integers consecutive in T belong both to T1or both to T2.T1 denotes the one containing b k if b k∈T.The four sets S X,S Y,T1,T2all have empty intersections.Hence,if all X i can take their value in S X∪T1and all Y j in S Y∪T2,we have a bound support for(X k,b k)on the Disjoint constraint.We have to prove that[min(X i)..max(X i)]intersects S X∪T1for any i∈{1,..,n} (and similarly for Y j and S Y∪T2).Since X i=Y j is BC for any j,min(X i)and max(X i)cannot be in S Y.If min(X i)or max(X i)is in S X or T1,we are done. Now,if both min(X i)and max(X i)are in T2,this means that there is a value between min(X i)and max(X i),which is in T1,by construction of T1and T2. As a result,any bound is BC on Disjoint if the decomposition is BC.♥From Theorem9,we deduce that BC can be achieved on Disjoint in polynomial time.In fact,we can achieve more than BC in polynomial time.Theorem10.AC on the binary decomposition is strictly stronger than BC on Disjoint.Proof.AC on the decomposition is at least as strong BC on the decompo-sition which is equivalent to BC on the original constraint.The following ex-ample shows strictness.Consider X1∈{1,2,3}and Y1∈{2}.The constraint Disjoint([X1],[Y1])is BC whereas GAC on the decomposition prunes2from X1.♥38 C.Bessiere et al.InLb:=f([lb(S1,..,lb(S n))],K);1InUb:=f([ub(S1,..,ub(S n))],K);2min(N):=max(min(N),InLb);3max(N):=min(max(N),InUb);4if min(N)>max(N)then fail;5if max(N)=InLb then6foreach S i.lb(S i)∩K=∅do ub(S i):=ub(S i)\K;if min(N)=InUb then7foreach S i.lb(S i)∩K=∅∧|K∩ub(S i)|=1do lb(S i):=lb(S i)∪K∩ub(S i);Among,Common and Disjoint Constraints39first assign all S i’s with their lower bound.InLb of the S i’s necessarily contain some k j since their lower bound overlaps K.For k−InLb variables among the InUb−InLb variables with some k j in their upper bound but none in their lower bound,we take some k j from their upper bound to obtain a satisfying assign-ment with N=k.Since min(N)≥InLb and max(N)≤InUb(lines3and4), N is BC.Suppose now a value v in ub(S i).The only case in which v should not belong to ub(S i)is when v is in the k j’s,none of the k j’s appear in lb(S i),and no more variable can take values in the k j’s,i.e.,InLb=max(N).Then,v will have been removed from ub(S i)(line6).In addition,suppose v should belong to lb(S i).This is the case only if there is no k j in lb(S i),and v is the only value in ub(S i)appearing in the k j’s,and InUb=min(N).Then,v will have been added in lb(S i)(line7).Computing the counters InLb and InUb is in O(nd).Updating the bounds on N is constant time.Deleting values that are not bound consistent in a ub(S i) or adding a value in lb(S i)is in O(d).Since there are n variables,this phase is again in O(nd).Bound consistency on Among is in O(nd).♥Note we can also add non-empty or cardinality conditions to the S i without making constraint propagation intractable.We again consider an extension in which we replace[k1,..,k m]by a set vari-able S.Unlike the previous Among constraint,enforcing BC on Among([S1,.., S n],S,N)is NP-hard in general.Theorem12.Enforcing BC on Among([S1,..,S n],S,N)is NP-hard. Proof.We reuse the reduction from the proof of Theorem2with minor mod-ifications.We create2k+1set variables for each literal i in the formula such that S i1..S ik∈{i}..{i},S i(k+1)..S i(2k)∈{−i}..{−i},and S i(2k+1)∈{}..{i,−i}. We create a set variable T j for each clause j inϕand let T j∈{}..{x,−y,z} where the j th clause inϕis x∨¬y∨z.We let N=n(k+1)+m and {}⊆S⊆{1,−1,..,n,−n}.The constraint Among([S11,..,S1(2k+1),..,S n1,.., S n(2k+1),T1,..,T m],S,N)has a solution iffϕhas a model.♥Note that the constraint remains intractable if the S i are non-empty or have afixed cardinality.We can easily modify the reduction by adding distinct “dummy”values to S i and T j respectively.We also add these dummy values to the lower bound of S.Despite this intractability result,we can easily modify Algorithm3to derive afiltering algorithm for Among([S1,..,S n],S,N)without changing the com-plexity.We use the lower bound of S(resp.ub(S))in the computation of InLb (resp.InUb).Also,instead of K in line6(resp.line7),we use lb(S)(resp. ub(S)).Finally,we need to consider the bounds of S.We remove v from ub(S) if|{i|lb(S i)⊆lb(S)∪{v}}|>max(N).Similarly,we add v to lb(S)if|{i| ub(S i)∩ub(S)\{v}=∅}|<min(N).We can easily extend the soundness proof of Theorem11for this algorithm as well.Such an algorithm does not achieve BC(otherwise we would have a polynomial algorithm for a NP-hard problem).Finally,the constraint Among([S1,..,S n],[T1,..,T m],N)is very similar to the previous one since several set variables[T1,...,T m]behave like their union.That40 C.Bessiere et al.S:=i∈{1,..,n}(lb(S i));1T:=i∈{1,..,m}(lb(T i));2if S∩T=∅then fail;3foreach S i do ub(S i):=ub(S i)\T; 4foreach T j do ub(T j):=ub(T j)\S; 5Among,Common and Disjoint Constraints41 First,if the lower bounds of the S i do not overlap those of the T j,then assigning all S i and T j to their lower bound is a solution.Thus,the lower bounds are BC. Now,for each value in the upper bound of each S i,we can construct a satisfying assignment involving v by assigning all other S i to their lower bounds,S i to its lower bound plus the element v,and all the T j to their corresponding lower bounds as none has v as an element.If d is the total number of values appearing in the upper bounds of the set variables,then at worst case the complexity of line4is O(nd)and of line5is O(md).Hence,the algorithm runs in O((n+m)d).♥Note that if we add a cardinality restriction to the size of the set variables,it becomes NP-hard to enforce BC on this constraint.5Related WorkAmong([X1,..,X n],[k1,..,k m],N)wasfirst introduced in CHIP by[3].A closely related constraint is the Count constraint[11].Count([X1,..,X n],v,op,N) where op∈{≤,≥,<,>,=,=}holds iffN op|{i|X i=v}|.The Among con-straint is more general as it counts the variables taking values from a set whereas Count counts those taking a given value.The algorithm of Among can easily be adapted to cover the operations considered in Count.There are other counting and related constraints for which there are specialised propagation algorithms such as Gcc[9],NValue[1],Same and UsedBy[4].In[6],a wide range of counting and occurrence constraints are specified using two primitive global constraints,Roots and Range.For instance,the Among on integer variables constraint is decomposed into a Roots and set cardinality constraint.Similarly,the Common constraint is decomposed into two Roots, two Range and two set cardinality constraints.However,Roots and Range cannot be used to express the Among,Common,and Disjoint constraints on set variables.Finally our approach to the study of the computational complexity of rea-soning with global constraints has been proposed in[5].In particular,as in[5], we show how computational complexity can be used to determine when a lesser level of local consistency should be enforced and when decomposing constraints will lose pruning.6ConclusionsWe have studied a number of variations of the Among,Common and Disjoint constraints over integer and set variables.Such constraints are useful in mod-elling problems involving resources.Our study shows that whether a global con-straint is tractable or not can be easily affected by a slight generalization or specialization of the constraint.However,a propagation algorithm for an in-tractable constraint can often be adapted from a propagation algorithm of a42 C.Bessiere et al.Table1.Summary of complexity results ConstraintInteger VariablesAmong([X1,..,X n],K,N)Among([X1,..,X n],S,N)Among([X1,..,X n],[Y1,..,Y m],N)Common(N,M,[X1,..,X n],[Y1,..,Y m])Disjoint([X1,..,X n],[Y1,..,Y m])Set VariablesAmong([S1,..,S n],K,N)Among([S1,..,S n],S,N)Among([S1,..,S n],[T1,..,T m],N)Common(N,M,[S1,..,S n],[T1,..,T m])Disjoint([S1,..,S n],[T1,..,T m])Among,Common and Disjoint Constraints43 References1.Beldiceanu,N.2001.Pruning for the minimum constraint family and for thenumber of distinct values constraint family.In Proc.of CP2001,211–224.Springer.2.Beldiceanu,N.2000.Global constraints as graph properties on a structurednetwork of elementary constraints of the same type.Technical report T2000/01, Swedish Institute of Computer Science.3.Beldiceanu,N.,and Contegean,E.1994.Introducing global constraints in CHIP.Mathematical Computer Modelling20(12):97–123.4.Beldiceanu,N.,Katriel,I.,and Thiel,S.2004.Filtering algorithms for the sameand usedby constraints.MPI Technical Report MPI-I-2004-1-001.5.Bessiere,C.,Hebrard,E.,Hnich,B.and Walsh,T.2004.The Complexity ofGlobal Constraints.In Proc.of AAAI2004.AAAI Press/The MIT Press.6.Bessiere,C.,Hebrard,E.,Hnich,B.,Kizilitan Z.and Walsh,T.2005.The Rangeand Roots Constraints:Specifying Counting and Occurrence Problems.In Proc.of IJCAI2005,60–65.Professional Book Center.7.Cheng,B.M.W.,Choi,K.M.F.,Lee,J.H.M.and Wu,J.C.K.1999.Increasing Con-straint Propagation by Redundant Modeling:an Experience Report.Constraints 4(2):167–192.8.R´e gin,J.-C.1994.Afiltering algorithm for constraints of difference in CSPs.InProc.of AAAI1994,362–367.AAAI Press.9.R´e gin,J.-C.1996.Generalized arc consistency for global cardinality constraints.In Proc.of AAAI1996,209–215.AAAI Press/The MIT Press.10.Sadler,A.,and Gervet,C.2001.Global reasoning on sets.In Proc.of Workshopon Modelling and Problem Formulation(FORMUL’01).Held alongside CP2001.11.Swedish Institue of Computer Science.2004.SICStus Prolog User’s Manual,Release 3.12.0.Available at http://www.sics.se/sicstus/docs/latest/pdf/ sicstus.pdf.。
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Galaxy Evolution across the Hubble Time Proceedings IAU Symposium No.235,bes &J.Palous,eds.c 2006International Astronomical Union DOI:00.0000/X000000000000000X Constraints on Bars in the Local Universe from 5000SDSS Galaxies Fabio D.Barazza,Shardha Jogee,and Irina Marinova Department of Astronomy,University of Texas at Austin,1University Station C1400,Asutin,TX 78712-0259,USA email:barazza@ Abstract.We present the first study of bars in the local Universe,based on the Sloan Digitized Sky Survey (SDSS).The large sample of ∼5000local galaxies provides the largest study to date of local bars and minimizes the effect of cosmic variance.The sample galaxies have M g −18.5mag and cover the redshift range 0.01 z <0.04.We use a color cut in the color-magnitude diagram and the S´e rsic index n to identify disk galaxies.We characterize bars and disks using r -band images and the method of iterative ellipse fits and quantitative criteria developed in Jogee at al.(2004,ApJL ,615,L105).After excluding highly inclined (i >60◦)systems our results are:(1)the optical (r -band)fraction of barred galaxies among local disk galaxies is 43%(Figure 1,left panel),which confirms the ubiquity of local bars,in agreement with other optical studies based on smaller samples (e.g.Eskridge et al.2000,AJ ,119,536,Marinova &Jogee 2006,astro-ph/0608039);(2)the optical bar fraction rises for bluer galaxies,suggesting a relation between bars and star formation (Figure 1,middle panel);(3)preliminary analyzes suggest that the optical bar fraction increases steeply with the galaxy effective radius (r ef f ,Figure 1,right panel);(4)the optical bar fraction at z ∼0is ∼35%for bright disks (M g −19.3mag)and strong (bar ellipticity >0.4),large-scale (bar semi-major axis >1.5kpc)bars,which is comparable to the value of ∼30±6%reported earlier (Jogee et al.2004)for similar disks and bars at z ∼0.2−1.
0.Figure 1.The optical bar fraction as a function of M g (left panel),g −r color (middle panel),and r ef f (right panel).The number next to each point denotes the total number of galaxies in
the corresponding bin.The dashed line indicates the total optical bar fraction.Only bins with more than 10objects are shown.
Keywords.galaxies:evolution,galaxies:formation,galaxies:
structure。